pipe under Hamiltonian System

Chinese

Chinese Journal of Aeronautics 21(2008) 35-42

Journal of Aeronautics

www.elsevier.com/locate/cja

Static/dynamic Analysis of Functionally Graded and Layered Magneto-electro-elastic Plate/pipe under Hamiltonian System Dai Haitao a,*, Cheng Weia, Li Mingzhib a

School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China b

Machinery and Electricity Engineering College, Shihezi University, Shihezi 832000, China Received 18 January 2007; accepted 20 September 2007

Abstract The 3-dimensional couple equations of magneto-electro-elastic structures are derived under Hamiltonian system based on the Hamilton principle. The problem of single sort of variables is converted into the problem of double sorts of variables, and the Hamilton canonical equations are established. The 3-dimensional problem of magneto-electro-elastic structure which is investigated in Euclidean space commonly is converted into symplectic system. At the same time the Lagrange system is converted into Hamiltonian system. As an example, the dynamic characteristics of the simply supported functionally graded magneto-electro-elastic material (FGMM) plate and pipe are investigated. Finally, the problem is solved by symplectic algorithm. The results show that the physical quantities of displacement, electric potential and magnetic potential etc. change continuously at the interfaces between layers under the transverse pressure while some other physical quantities such as the stress, electric and magnetic displacement are not continuous. The dynamic stiffness is increased by the piezoelectric effect while decreased by the piezomagnetic effect. Keywords: functionally graded magneto-electro-elastic material; Hamiltonian system; symplectic algorithm

1 Introduction* Functionally graded magneto-electro-elastic materials (FGMM) which acting as the functional material in smart structures and sensors have attracted many attentions in both theoretical and engineering areas. Various studies have been carried out about the static and dynamic characteristics and many theoretical and numerical solutions are derived. The most popular numerical approaches for solving the structural stability of multilayered FGMM system are the state-vector[1-5] and transfer matrix[6-13] methods. The general displacement and

*Corresponding author. Tel.: +86-10-81912191. E-mail address: [email protected] Foundation item: Aeronautical Science Foundation of China (20071551016)

stress of the medium are divided into so-called “out-of-plane variables” and “in-plane variables”. Then the state-vector equations of the “out-of-planevariables” are established by means of variableelimination of the governing equations, which is a very complicated process and quite variant for different materials and structures. In this paper, the 3-dimensional couple equations of magneto-electroelastic structures are derived under Hamiltonian system based on the Hamilton principle. The problem of single sort of variables is converted into the problem of double sorts of variables, and the Hamilton canonical equations are established. Although the Hamilton canonical equations and the statevector equations are exactly the same, the method-

· 36 ·

Dai Haitao et al. / Chinese Journal of Aeronautics 21(2008) 35-42

ology of this paper is based on variational principle which is more rational and universal. Then the equations are solved by symplectic algorithm which has been proved to be a precise and robust algorithm[14-15].

2.1 Governing equation Geometric equations (1)

The antithetic vector can be expressed as p=

∂L ⎡ ∂L =⎢ ∂q ⎣ ∂w1

∂L ∂w2

∂L ∂w3

∂L ⎤ ⎥ ∂ψ ⎦

∂L ∂φ

T

(4)

(5)

According to the Hamilton principle[16]: δΠ g = δ{∫∫∫ ⎡⎣ p T q − H ( p, q ) ⎤⎦ dΩ − ∫∫ n jσ ij ui dS − Ω Su

∫∫S

(n jσ ij − f i )ui dS − ∫∫ ni DiΦ dS − ∫∫ (ni Di − g )i SΦ

σ

(2)

where u, σij and sij are the displacement, stress and strain; φ, Di and Ek are the electric potential, electric displacement and electric intensity; ψ, Bi and Hk are the magnetic potential, magnetic induction intensity and magnetic intensity; cijkl, εik and μik are the elastic modulus, dielectric coefficient and magneto-electric coefficient; and eijk, dik and qijk are the piezoelectric coefficient, magnetic permeability coefficient and piezomagnetic coefficient respectively. All the subscripts i, j, k, l = 1, 2, 3 or x, y, z. 2.2 Hamiltonian system of magneto-electroelastic structures The energy of magneto-electro-elastic dynamic system includes the following parts: kinetic energy T, elastic potential energy V and electromagnetic energy W. The Lagrange energy density can be expressed as follows L = T −V +W

T

Η ( p, q ) = p T q ( q , p ) − L ( q , p )

Constitutive equations

σ ij = cijkl skl − eijk Ek − qijk H k ⎫ ⎪⎪ Di = eikl skl + ε ik Ek + dik H k ⎬ Bi = qijk s jk + dik Ek + μik H k ⎪ ⎪⎭

u3 φ ψ ]

where x = ∂ (i) / ∂z and ∂ (i) / ∂r in Cartesian coordinates and polar coordinates respectively. Hamiltonian function of compound energy can be expressed as

2 Computation Scheme

sij = (ui , j + u j ,i ) / 2 ⎫ ⎪ Ei = −φ,i ⎬ ⎪ H i = −ψ ,i ⎭

q = [u1 u2

(3)

1 1 1 where V = σ ij sij , T = ρ u'i u'i , W = Ei Di + 2 2 2 1 H i Bi . 2 Generalized displacement is defined as

SD

Φ dS − ∫∫ ni Biψ dS − ∫∫ (ni Bi − m)ψ dS } = 0 SΦ

SD

(6) Then the Hamilton canonical equations can be established as follows ∂H ⎫ ∂p ⎪⎪ ⎬ ∂H ⎪ p (q , p ) = − ∂q ⎪⎭

q (q , p ) =

(7)

Eq.(7) can be simplified as follows when statevector v is introduced: dν = Hν dz

(8)

where v = [qT pT]T, H is named as Hamiltonian operator matrix. H can be expressed as follows in Cartesian coordinates ⎡ A −D ⎤ H =⎢ ⎥ ⎣ − B − A' ⎦

where ⎡ ⎢ 0 ⎢ ⎢ ⎢ 0 A=⎢ ⎢ −χ D ⎢ 11 x ⎢ − χ 21 Dx ⎢ ⎣⎢ − χ 31 Dx

e15 Dx C55

0

− Dx

−

0

− Dy

−

− χ12 D y

0

0

− χ 22 D y

0

0

− χ 32 D y

0

0

e24 D y C44

q15 ⎤ Dx ⎥ C55 ⎥ ⎥ q24 Dy ⎥ − C44 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎦⎥ −

Dai Haitao et al. / Chinese Journal of Aeronautics 21(2008) 35-42

0 χ11 Dx χ 21 Dx χ 31 Dx ⎤ ⎡ 0 ⎢ 0 0 χ12 Dy χ 22 Dy χ 32 Dy ⎥⎥ ⎢ ⎢ D Dy 0 0 0 ⎥ x ⎢ ⎥ A' = ⎢ e D ⎥, e D 24 y 15 x 0 0 0 ⎥ ⎢ C44 ⎢ C55 ⎥ ⎢q ⎥ q 24 ⎢ 15 Dx 0 0 0 ⎥ Dy C44 ⎢⎣ C55 ⎥⎦ ⎡α11 Dx2 + C66 Dy2 − ρ Dt2 ⎢ ⎢ (α12 + C66 ) Dx Dy α 22 Dy2 + C66 Dx2 − ρ Dt2 ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ ⎢ 0 0 B=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ ⎣

⎡ 1 ⎢− C ⎢ 55 ⎢ ⎢ 0 D=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎣

− ρ Dt2

0

0

⎡C13 χ = ζγ , γ = ⎢⎢ e31 ⎢⎣ q31 ⎡C33 −1 ζ = ξ , ξ = ⎢⎢ e33 ⎢⎣ q33

−ε 33 −d33

sym ⎤ ⎥ ⎥ − μ33 ⎥⎦

sym

0

−ζ 11

0

−ζ 21

−ζ 22

0

−ζ 31

−ζ 32

⎡H H = ⎢ 11 ⎣ H 21

H12 ⎤ ⎥ H 22 ⎦

H11 = ⎡ ˆ ⎢ −C12 ⎢ ⎢ − Dθ ⎢ ⎢ −rDz ⎢ 0 ⎢ ⎢⎣ 0

C23 ⎤ e32 ⎥⎥ q32 ⎥⎦

1 C44

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ −ζ 33 ⎥⎦

⎤ ⎥ ⎥ ⎥ ⎥ sym ⎥ 2 e15 ⎥ 2 −(ε11 + ) Dx − ⎥ C55 ⎥ 2 ⎥ e24 2 (ε 22 + ) Dy ⎥ C44 ⎥ ⎥ 2 q −( μ11 + 15 ) Dx2 − ⎥ ⎥ C55 0 ⎥ 2 q24 ⎥ 2 ( μ22 + ) Dy ⎥ C44 ⎦

where DΔ = ∂ ∂Δ , “sym” means the symmetry of matrix, A' is the mutual-adjoint matrix of A.

α ij = Cij − Ci 3 χ1 j − e3i χ 2 j − q3i χ3 j (i = 1, 2; j = 1, 2)

−

· 37 ·

In the polar coordinates, for the convenience of analysis a new state-vector is defined as v = [qT rpT]T. So the Hamilton canonical equations are converted into follows dν = Hν (9) dr where

re31 Dz C11

−Cˆ12 Dθ

− rCˆ13 Dz

1

0

0

0 0 0

0 0 0

0 0 0

H12

⎡ 1 ⎢C ⎢ 11 ⎢ ⎢ 0 =⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎣

−

1 C66 0 0 0

ζ 11 ζ 12 ζ 13

−

rq31 ⎤ Dz ⎥ C11 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥⎦

⎤ ⎥ ⎥ ⎥ sym ⎥ ⎥ ⎥ ⎥ ζ 22 ⎥ ζ 23 ζ 33 ⎥⎦

H 21 = ⎡ r 2 ρ Dt2 + Q22 ⎢ ⎢ −Q22 Dθ ⎢ ⎢ −rQ23 Dz ⎢ −rR D 11 z ⎢ ⎢ −rR D 21 z ⎣

Q22 Dθ r

2

ρ Dt2

2

− Q22 Dθ − r

rQ23 Dz 2

rR11 Dz

C44 Dz2

−r (C44 + Q23 ) Dθ Dz

r 2 ρ Dt2 − C44 Dθ2 − r 2Q33 Dz2

−r (e24 + R11 ) Dθ Dz

− r 2 R12 Dz2 − e24 Dθ2

r 2 K11 Dz2 + ε 22 Dθ2

−r (q24 + R21 ) Dθ Dz

− r 2 R22 Dz2 − q24 Dθ2

r 2 K 21 Dz2 + d 22 Dθ2

⎤ ⎥ ⎥ ⎥ sym ⎥ ⎥ ⎥ r 2 K 22 Dz2 + μ22 Dθ2 ⎦⎥ rR21 Dz

· 38 ·

Dai Haitao et al. / Chinese Journal of Aeronautics 21(2008) 35-42

⎡ Cˆ12 ⎢ ⎢ −Cˆ12 Dθ = ⎢ −rCˆ D ⎢ 13 z ⎢ − reˆ D 31 z ⎢ ⎢⎣ − rqˆ31 Dz

− Dθ

−rDz

−1

0

0

0

0

0

0

0

⎡C55 −1 ζ = (ξ ) , ξ = ⎢⎢ e15 ⎣⎢ q15

−ε11

H 22

− d11

0 0⎤ ⎥ 0 0⎥ ⎥ 0 0⎥ 0 0⎥ ⎥ 0 0 ⎥⎦ sym ⎤ ⎥ ⎥ − μ11 ⎦⎥

2 K11 = ε 33 + e31 C11 , K 21 = d33 + e31q31 C11

K 22 = C11 , R11 = e32 − Cˆ12 e31 R12 = e33 − Cˆ13e31 , R21 = q32 − Cˆ12 q31 R22 = q33 − Cˆ13q31 , eˆij = eij C11 , qˆij = qij C11 2 μ33 + q31

Cˆij = Cij C11

, Qij = Cij − C1iC1 j C11

2.3 Solution of the Hamilton canonical equations For the case of FGMM plate, the general displacement vector and its antithetic vector are expressed as follows qt = ⎡⎣u x pt = ⎡⎣σ xz

uy

uz

φ ψ ⎤⎦

σ yz σ zz

Dz

pressed as follows qp = [ur pp = [σ rr

uθ

uz

σ rθ

φ ψ]

T

σ rz

Dr

Br ]

T

Consider a simply supported FGMM pipe of length l, inner radius R and thickness hp, the dimensionless vector in Hamilton canonical equations can be expanded as uθ h p ⎧ ⎫ ⎪ ⎪ uz h p ⎛ uθ (ς ) sin(mπλ ) sin(nθ ) ⎞ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ u z (ς ) cos(mπλ ) cos(nθ ) ⎟ ur h p ⎜ ⎪ ⎪ ⎜ ur (ς ) sin(mπλ ) cos(nθ ) ⎟ ⎪ 0 0 ⎪ ⎜ ⎟ ⎪ φ h p c44 ε 33 ⎪ ⎜ φ (ς ) cos(mπλ ) cos(nθ ) ⎟ ⎪ ⎪ 0 0 ⎪ψ h p c44 μ33 ⎪ ∞ ∞ ⎜ ψ (ς ) cos(mπλ ) cos(nθ ) ⎟ iωt ⎟e ⎨ ⎬ = ∑ ∑⎜ 0 σ rz c44 ⎪ ⎪ m =1 n =1 ⎜ σ rz (ς ) cos(mπλ ) cos(nθ ) ⎟ ⎜ σ (ς ) sin(mπλ ) sin(nθ ) ⎟ ⎪ ⎪ 0 σ rθ c44 ⎜ rθ ⎟ ⎪ ⎪ ⎜ σ rr (ς ) sin(mπλ ) cos(nθ ) ⎟ ⎪ ⎪ 0 σ rr c44 ⎜ ⎟ ⎪ ⎪ ⎜ Dr (ς ) cos(mπλ ) cos(nθ ) ⎟ ⎪ ⎪ 0 0 ⎜ B (ς ) cos(mπλ ) cos(nθ ) ⎟ ⎪ Dr c44ε 33 ⎪ ⎝ r ⎠ ⎪ ⎪ 0 0 ⎩ Br c44 μ33 ⎭

T

(11) Bz ⎤⎦

T

Consider a simply supported rectangular FGMM plate of length a, width b and thickness ht, the dimensionless vector in Hamilton canonical equations can be expanded as ux h t ⎧ ⎫ ⎪ ⎪ ⎛ u x (γ ) cos(mπα )sin(nπβ ) ⎞ uy h t ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ uz h t ⎜ u y (γ ) sin(mπα ) cos(nπβ ) ⎟ ⎪ ⎪ ⎜ u (γ )sin(mπα )sin(nπβ ) ⎟ ⎪ 0 0 ⎪ ⎜ z ⎟ ⎪ φ h t c44 ε 33 ⎪ ⎜ φ (γ )sin(mπα ) sin(nπβ ) ⎟ ⎪ ⎪ 0 0 ⎜ ⎟ ⎪⎪ψ h t c44 μ33 ⎪⎪ ∞ ∞ ⎜ ψ (γ ) sin(mπα )sin(nπβ ) ⎟ iωt = ∑∑ e ⎨ ⎬ 0 ⎪ σ xz c44 ⎪ m =1 n =1 ⎜ σ xz (γ ) cos(mπα )sin(nπβ ) ⎟ ⎜ ⎟ ⎪ ⎪ 0 ⎜ σ yz (γ ) sin(mπα ) cos(nπβ ) ⎟ ⎪ σ yz c44 ⎪ ⎜ σ (γ ) sin(mπα ) sin(nπβ ) ⎟ ⎪ ⎪ 0 ⎜ zz ⎟ ⎪ σ zz c44 ⎪ π π ( ) sin( ) sin( ) D γ m α n β ⎜ ⎟ z ⎪ ⎪ 0 0 ⎜ ⎟ ⎪ Dz c44ε 33 ⎪ π π ( )sin( ) sin( ) B γ m α n β ⎝ z ⎠ ⎪ ⎪ 0 0 ⎪⎩ Bz c44 μ33 ⎪⎭ (10)

where α = x/a, β = y/b and γ = z/ht. For the case of FGMM pipe, the general displacement vector and its antithetic vector are ex-

Where λ = z/l, ς = r/hp. Substituting Eq.(10) into Eq.(8) and Eq.(11) into Eq.(9) respectively, the arbitrary couple of (m, n) is obtained: dνmn = H mn νmn dr

(12)

The recursion formula of the above equation is v (ri +1 ) = exp( H i Δr )v (ri )

(13)

where ri and ri+1 are the coordinates along the depth of the pipe wall. Δr = ri+1 – ri, exp(HiΔr ) is the transfer matrix of these two sections and it is calculated by symplectic algorithm. Using Eq.(13) repeatedly then the relationship between the two statevectors on both surfaces is established. v ( R2 ) = Κv ( R1 )

(14)

∞

where Κ = ∏ exp( H i Δr ) . i =1

3 Presentation of Result As mentioned in Ref.[9], there are two main

Dai Haitao et al. / Chinese Journal of Aeronautics 21(2008) 35-42

models of functionally graded material. For the homogeneous material, the property along the depth can be expressed as

ξ h (h ) = ξ exp(κ h )

of two layers and all of them change intensively as the exponential factor increases.

(15)

where ξh and h represent the material property and dimensionless depth respectively, κ is the exponential factor governing the gradient of the material. For the non-homogeneous material the property along the depth can be expressed as follows Fig.1

ξ n = ξ1V1 + ξ 2V2 ⎫ η

V1 = ( z hn ) V2 = 1 − V1

⎪⎪ ⎬ ⎪ ⎪⎭

(16)

where Vi and ξi are the volume proportion and property of material i (i = 1,2), η is the exponential factor (η ≥ 0). 3.1 Static analysis of simply supported multilayered FGMM plate A simply supported multilayered FGMM plate with stacking sequences B/F/B (B and F represent BaTiO3 and CoFe2O4 respectively) is subjected to a distributed load on the top surface, as shown in Fig.1. The material properties have been introduced in Ref.[9]. Both outer layers are graded BaTiO3 while the inner layer is homogenous CoFe2O4. The responses are calculated for fixed horizontal coordinates (x, y)=(0.37a, 0.27b) where a = b = 1 m, h1 = h2 = h3 = 0.01 m and shown in Fig.2. Fig.2(a)-(g) show the variations of the out-ofplane variables and Fig.2(h)-(k) show the variations of the in-plane variables of the FGMM plate. It can be seen that the out-of-plane variables are continuous across the thickness while the in-plane variables are discontinuous. As exponential factor κ increases the magnitude of dielectric displacement decreases while that of magnetic induction increases. In addition, the slope of their magnitudes changes intensively. The vertical displacement changes very slightly along the thickness and is not similar to that of the electric and magnetic potential. For the case of in-plane variables, there is a jump at the interface

· 39 ·

Sketches of three-layer FGMM plate.

· 40 ·

Dai Haitao et al. / Chinese Journal of Aeronautics 21(2008) 35-42

Fig.2

Static responses of layered FGMM plate under mechanical load.

Dai Haitao et al. / Chinese Journal of Aeronautics 21(2008) 35-42

· 41 ·

3.2 Modal analysis of simply supported layered FGMM plate

3.3 Modal analysis of simply supported layered FGMM pipe

In order to analyse the effect of electric and magnetic coupling on the dynamic character of FGMM structures, the following five different classes mentioned in Ref.[9] are discussed: ① considering elastic property of the structure only; ② accounting the coupling among magnetoelectro-elastic field; ③ neglecting magneto-electric coupling; ④ considering magnetostrictive field only; ⑤ considering piezoelectric field only. A square plate of ht = h1 + h2 + h3 = a = b = 0.3 m studied by W.Q. Chen[6] is analyzed for comparison. The natural frequencies are listed in Table 1 with the results of this paper (upper lines) and Chen’s results (lower lines), and Kt is the modal order along the depth. It shows that both results are reasonable identical. Some Chen’s missed frequencies are validated here.

A FGMM hollow cylinder, its inner surface is pure BaTiO3 and outer surface is pure CoFe2O4. The geometry of the pipe is r = 0.7 m, l = 4 m, hp = 0.6 m.

Table 1

Natural frequencies of three-layer FGMM plate ( Ω = ω a ρ max cmax )

Fig.3

The comparison of the vibration frequencies among four kinds of states is presented in Table 2 in which the lower lines are the results in Ref.[9]. Both results are quite consistent with each other. Table 2 η

Comparison among four states (Unit: Hz)

m,n,Kp 1,1,1

Kt

B/F/B

Sketches of FGMM pipe.

F/B/F 1,2,1

Ⅰ

Ⅱ

Ⅰ

Ⅱ

1

1.477 –

1.547 1.547

1.530 –

1.605 1.605

2

1.824 1.824

1.824 –

1.898 1.898

1.898 –

1,4,1

3

2.155 –

2.155 2.155

2.105 –

2.247 2.249

1,5,1

4

2.156 –

2.244 –

2.315 2.315

2.315 –

1,1,1

5

2.924 –

3.076 –

3.115 3.115

3.115 –

6

3.076 3.076

3.083 3.084

3.159 –

3.221 3.227

7

3.331 –

3.443 3.444

3.711 –

3.736 3.743

8

4.114 4.115

4.114 –

4.176 4.177

4.176 –

9

4.128 –

4.391 4.393

4.385 –

4.397 4.398

10

4.566 –

4.697 –

4.453 –

4.531 –

11

5.246 5.256

5.246 –

5.307 5.309

5.307 –

0

1,3,1

1,2,1 1 000

1,3,1 1,4,1 1,5,1

I

II

IV

V

201.55 201.88 348.76 349.35 758.10 759.38 1 207.74 1 209.78 1 659.05 1 661.83

202.03 202.36 349.06 349.65 758.93 760.21 1 208.80 1 210.84 1 660.19 1 662.98

201.55 201.01 348.76 349.57 758.10 759.86 1 207.74 1 210.55 1 659.05 1 662.89

202.03 202.49 349.06 349.87 758.93 760.69 1 208.80 1 211.61 1 660.19 1 664.04

216.78

216.19

216.19

216.70

217.30

216.88

216.82

217.40

401.56

401.55

401.55

401.50

402.25

402.24

402.49

402.50

873.59

873.56

873.56

873.50

874.97

874.93

875.49

875.50

1 386.42

1 386.31

1 386.31

1 386.42

1 388.59

1 388.48

1 389.36

1389.48

1 897.37

1 897.15

1 897.15

1 897.37

1 900.32

1 900.10

1 901.30

1 901.54

The natural frequencies of case II are little less than those of case V which indicates that the piezomagnetic effect decreases the dynamic stiffness of FGMM structure. The natural frequencies of case I are less than those of case V which indicates that the

· 42 ·

Dai Haitao et al. / Chinese Journal of Aeronautics 21(2008) 35-42

piezoelectric effect increases the dynamic stiffness of FGMM structure.

4 Conclusions (1) The state-vector theory of piezoelectric and piezomagnetic structure are constructed based on electromagnetic elastic solid variational principles and Legendre transformation. Although the Hamilton canonical equations and the state-vector equations are exactly the same, the methodology used in this paper is based on variational principle which is more rational and universal. (2) The results show that the physical quantities of displacement, electric potential and magnetic potential etc. change continuously at the interfaces under the transverse pressure while some other physical quantities such as the stress, electric and magnetic displacement are not continuous. (3) The higher-order modals of FGMM plate and pipe are obtained and their accuracy is validated through comparison with the results of other papers. It shows that the dynamic stiffness is increased due to the piezoelectric effect while decreased due to the piezomagnetic effect. References [1]

geneous transversely isotropic magneto-electro-elastic plates. Journal of Sound and Vibration 2005; 279(1-2): 237-251. [7]

on anti-plane problem of magneto-electro-elastic solids. Journal of Dalian University of Technology 2004; 44(5): 630-633. [in Chinese] [8]

Structures 2007; 44(3-4): 1073-1085. [9]

Bhangale R K, Ganesan N. Free vibration studies of simply supported non-homogeneous functionally graded magneto-electroelastic finite cylindrical shells. Journal of Sound and Vibration 2005; 288(1-2): 412-422.

[10]

Chen W Q, Bian Z G, Lu C F, et al. 3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid. International Journal of Solids and Structures 2004; 41(3-4): 947-964.

[11]

Pan E, Han F. Exact solution for functionally graded and layered magneto-electro-elastic plates. International Journal of Engineering Science 2005; 43(3-4): 321-339.

[12]

Tarn J Q, Wang Y M. Laminated composite tubes under extension, torsion, bending, shearing and pressuring: a state space approach. International Journal of Solids and Structures 2001; 38(50-51): 9053-9075.

[13]

Senthil S V, Batra R C. Three-dimensional exact solution for the vibration of functionally graded rectangular plates. Journal of Sound and Vibration 2004; 272(3-5): 703-730.

[14]

Feng K. Difference schemes for Hamiltonian formulism and sym-

27(9): 1028-1032. [in Chinese]

plectic geometry. Journal of Computational Mechanics 1986; 4(3):

Guan Q, He S R. Two-dimensional analysis of piezoelectric/

279-289.

Guan Q, He S R. Three-dimensional analysis of piezoelectric/

[15]

Zhong W X. Duality system of applied mechanics. Beijing: Science Press, 2002. [in Chinese]

[16]

Yao W A, Li X C. Symplectic duality system on the plane mag-

piezomagnetic elastic media. Composite Structures 2006; 72(4):

neto-electro-elastic solids. Applied Mathematics and Mechanics

419-428.

2006; 27(2): 177-185. [in Chinese]

Wang J G, Chen L F, Fang S S. State vector approach to analysis of multilayered magneto-electro-elastic plates. International Journal of Solids and Structures 2003; 40(7): 1669-1680.

[5]

Chen J Y, Pan E, Chen H L. Wave propagation in magneto-electro-elastic multilayered plates. International Journal of Solids and

69(2): 229-237.

[4]

Yao W A. Symplectic solution system and Saint-Venant principle

calculation of the state vectors in the multi-layered magneto-elec-

piezomagnetic and elastic media. Composite Structures 2005;

[3]

Chen W Q, Kang Y L, Ding H J. On free vibration of non-homo-

Zhu B B, Wang J G, Li G Q. A kind of selective algorithm for the

tro-elastic media. Journal of Hefei University of Technology 2004;

[2]

[6]

Wang J G, Fang S S, Chen L F. The state vector methods for space axisymmetric problems in multilayered piezoelectric media. International Journal of Solids and Structures 2002; 39(15): 39593970.

Biography: Dai Haitao Born in 1979, he is a Ph.D. candidate of Beijing University of Aeronautic and Astronautic. His main research interests are structure dynamic and smart structure. E-mail: [email protected]