Free vibration analysis of functionally graded thin annular sector plates using the differential quadrature method

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Free vibration analysis of functionally graded thin annular sector plates using the differential quadrature method

S H Mirtalaie^1∗andM A Hajabasi²

1Department of Mechanical Engineering, Islamic Azad University, Najafabad Branch, Najafabad, Iran

2Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

The manuscript was received on 20 January 2010 and was accepted after revision for publication on 28 July 2010.

DOI: 10.1243/09544062JMES2232

Abstract: In this article, the differential quadrature method (DQM) is used to study the free vibration of functionally graded (FG) thin annular sector plates. The material properties of the FG-plate are assumed to vary continuously through the thickness, according to the power-law distribution. The governing differential equations of motion are derived based on the classical plate theory and solved numerically using DQM. The natural frequencies of thin FG annular sector plates under various combinations of clamped, free, and simply supported boundary conditions are presented for the first time. To ensure the accuracy of the method, the natural frequencies of a pure metallic plate are calculated and compared with those existing in the literature for the homogeneous plate. In this case, the result shows very good agreement. For the FG-plates, the effects of boundary conditions, volume fraction exponent, and variation of Poisson’s ratio on the free vibrational behaviour of the plate are studied.

Keywords: free vibration analysis, functionally graded material, thin annular sector plate, differential quadrature method

1 INTRODUCTION

In recent years, the new microscopic inhomogeneous composites that are named functionally graded materials (FGMs) have attracted extensive attention in many fields of engineering. These materials are usually made of a combination of ceramic and metal such that the material properties vary smoothly and continuously in appropriate direction(s) [1]. The continuity in the material properties of these new types of composites provides better mechanical behaviour in comparison to the fibre-reinforced composites. The mismatch of material properties across the interface of two dis- crete materials in fibre-reinforced composites causes many deficiencies such as delamination, debonding, cracking, stress concentration, and residual stresses.

These phenomena can be reduced by gradation in properties of the materials in the FGMs. Moreover,

∗Corresponding author: Department of Mechanical Engineering, Islamic Azad University, Najafabad Branch, Najafabad, Iran.

email: [email protected]

a combination of the properties of the metal and ceramic can be achieved by the composition of them.

In other words, ceramics have wear, oxidation and high temperature resistance that the latter is due to their low thermal conductivity. Also, metals have properties such as the high toughness, high strength, machinabil- ity and bonding capability. Therefore, the composition of the metals and ceramics makes FGMs resistant to high-temperature conditions while their toughness is maintained. Because of these good characteristics, FGMs have been extensively used in various industries such as space structures, turbo-machinery, nuclear and chemical industries, defence mechanisms, energy conversion systems [2], tribology [3], systems in vibration and acoustic controls or condition monitoring [4], and semiconductor devices [5]. Due to this widespread applicability, FGMs have been extensively studied by researchers in recent years, particularly the bending and vibration analyses of functionally graded (FG) structures such as plates are carried out by many researchers [6–13]. As known by the authors, the vibration problem of annular sector-shaped plates made of functionally graded materials are less regarded so

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far, whereas components with sectorial geometries are extensively used in engineering structures such as aeronautical and naval structures, nuclear reac- tors, curved bridge decks, and panels. Therefore, the free vibration analysis of sector plates is of practi- cal importance in the structural mechanics and there are many articles in the literature that have been devoted to this topic [14–17]. Recently, sector and annular sector plates made of FGM are studied by a few researchers; Jomehzadehet al.[18] presented an exact analytical approach for the bending analysis of FG annular sector plates based on the first-order shear deformation plate theory. Sahraee [19] carried out the bending analysis of FG thick circular sector plates based on the Levinson plate theory and the first-order shear deformation plate theory. Nie and Zhong [20]

studied the three-dimensional (3D) free and forced vibration of FG annular sectorial plates with simply supported radial edges and arbitrary circular edges using a semianalytical approach.

More recently, Hosseini-Hashemiet al.[21] investi- gated the buckling and free vibration behaviours of radially FG circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. They considered the inhomogeneity of the plate according to the exponential variation of Young’s modulus and mass density of the material along the radial direction, whereas the Poisson’s ratio was assumed to be constant. In many applications of FGMs, the material properties vary continuously through the thickness of the plate. In these cases, due to the variation of elastic properties through the thickness, the in-plane and transverse motions of the plate are coupled together and the governing equations conclude three coupled partial differential equations (PDEs) which can hardly be solved exactly under various boundary conditions (BCs).

In this article, the differential quadrature method (DQM) is used for free vibration analysis of FG thin annular sector plates in which the material properties of the FGM consisting of Young’s modulus, mass density, and Poisson’s ratio are assumed to vary continuously through the thickness, according to the power-law distribution. The DQM that was introduced by Bellman and Casti [22] is a powerful method for solving initial and boundary value problems that need less computational efforts compared to the other numerical methods such as finite-element and finite- difference methods. The method was first used by Bertet al.[23] for solving problems in the structural mechanics and then has been widely used for static and free vibration analysis of beams and plates in various problems. Its early development and some of its applications can be found in review articles by Bert and Malik [24,25].

This article has the following structure. First, the governing equations of motion of the thin FG annular

sector plates and the corresponding BCs are derived according to the classical plate theory. Then the DQM is used to discretize the equations of motion as well as the BCs. Using this procedure, an eigenvalue problem is obtained that its solution represents the natural frequencies and mode shapes of the plate. As a particular case, the FG-plate can be converted to the homogeneous one. The results for the latter are presented here and compared with the available literature to demon- strate the validity of the present work. Then the results for FG-plates with various BCs consist of free, clamped, and simple supported are presented.

2 GOVERNING EQUATIONS

The differential equations of motion for free vibration of the FG thin sector plates can be derived using Hamilton’s principle. The Young’s modulus of elastic- ity E(z), Poisson’s ratioν(z), and the density ρ(z)of the FG annular plate is considered to vary in the thickness of the plate according to the following polynomial distributions [26]

E =E(z)=Em+(Ec−Em) 1

2+z h

kE

(1) ν=ν(z)=ν_m+(ν_c−ν_m)

1 2+z

h kν

(2) ρ=ρ(z)=ρ_m+(ρ_c−ρ_m)

1 2+ z

h kρ

(3) where the subscripts c and m refer to the ceramic and metal phases, respectively. Also, his the thickness of the plate in thezdirection andkis named the power law or volume fraction exponent of FGM.

Abrate [27] showed that because of the variation of the elastic properties through the thickness, the equations of motion governing the in-plane and transverse deformations are coupled, that is, both in-plane and transverse displacements must be taking into account in the analysis of FG-plates. Thus, in the cylindrical co-ordinate system, the displacement components in the r,θ, and z directions are defined asu,˜ υ˜, andw,˜ respectively. Based on the Kirchhoff hypothesis [28], the displacement field due to the bending and in-plane stretching are assumed to take the following form

˜

u(r,θ,z,t)=u(r,θ,t)−z∂w(r,θ,t)

∂r (4)

˜

υ(r,θ,z,t)=υ(r,θ,t)−z r

∂w(r,θ,t)

∂θ (5)

˜

w(r,θ,z,t)=w(r,θ,t) (6) whereu(r,θ,t),υ(r,θ,t), andw(r,θ,t)are the radial, circumferential, and transverse displacements of the mid-plane(z=0)of the plate at timet. The geometry

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Fig. 1 Geometry of the plate and co-ordinate system of the plate, dimensions, and the co-ordinate system are shown in Fig. 1.

Based on the above displacement components, the linear strain components of the elastic material are defined as [29–31]

{ε} =

⎧⎨

⎩ ε_rr ε_{θ θ} γ_rθ

⎫⎬

⎭=

⎧⎪

⎨

⎪⎩

ε⁰_rr+zκrr

ε⁰_{θ θ}+zκ_{θ θ} γ_rθ⁰ +zκ_rθ

⎫⎪

⎬

⎪⎭ (7)

in which the middle-plane strain vector{ε⁰}and the curvature vector{κ}are given by

ε⁰

=

⎧⎪

⎨

⎪⎩ ε⁰_rr ε⁰_{θ θ} γ_rθ⁰

⎫⎪

⎬

⎪⎭=

⎧⎪

⎪⎨

⎪⎪

⎩

u,r

υ_,θ r +u u,θ r

r +υ_,r−υ r

⎫⎪

⎪⎬

⎪⎪

⎭ ,

{κ} =

⎧⎨

⎩ κ_rr κ_{θ θ} κ_rθ

⎫⎬

⎭=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

w_,r w,θ θ

r² +^w_r^,r 2

w_,θ r² −w_,rθ

r

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

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where the subscripts denote differentiation with respect to the independent variables. Considering the material properties (1) and (2), the 2D stress–strain law for plane stress state can be written as

⎧⎨

⎩ σ_rr σ_{θ θ} σ_rθ

⎫⎬

⎭= [C]

⎧⎨

⎩ ε_rr ε_{θ θ} ε_rθ

⎫⎬

⎭,

[C] = E(z) 1−ν(z)²

⎡

⎢⎣

1 ν(z) 0

ν(z) 1 0

0 0 1−ν(z) 2

⎤

⎥⎦ (9)

In order to derive the governing differential equations of motion, Hamilton’s principle is used that states

_t₂

t1

(δΠ−δK)dt=0 (10)

in which the virtual strain energy δΠ of the plate is given by

δΠ=1 2

A

h/2

−h/2

(σ_rrδε_rr+σ_{θ θ}δε_{θ θ}+σ_rθδγ_rθ)dzdA (11) Considering the following resultant in-plane forces Nijand bending momentsMijas

N_ij,M_ij

= _h/2

−h/2

(1,z) σ_ijdz (12) where i and j stand for r and θ, respectively. Sub- stituting the strain components from equation (8) in equation (11) yields

δΠ=1 2

A

Nrr

∂δu

∂r −Mrr

∂²δw

∂r² +Nθ θ

δu r

−Mθ θ

r

∂δw

∂r +Nθ θ

r

∂δυ

∂θ −Mθ θ

r²

∂²δw

∂θ² +Nrθ

1 r

∂δu

∂θ −2Mrθ

r

∂²δw

∂r∂θ +Nrθ

∂δυ

∂r +2Mrθ

r²

∂δw

∂θ −Nrθ

δυ r +Q_r∂δw

∂r +Qθ

r

∂δw

∂θ −Q_r∂δw

∂r

− Qθ

r

∂δw

∂θ

dA (13)

Also the variation of kinetic energyδKof the plate is given by

δK =1 2

A

_h/2

−h/2

ρ(z) δ

× ∂u˜

∂t 2

+ ∂υ˜

∂t 2

+ ∂w˜

∂t 2

dZdA (14) Introducing the following inertia terms

Ii = h/2

−h/2zⁱρ(z)dz, i=0, 1, 2 (15) and the replacement of equations (4) to (6) into equation (14) yields

δK =

A

I0

∂u

∂t

∂δu

∂t +∂υ

∂t

∂δυ

∂t +∂w

∂t

∂δw

∂t

−I₁ ∂²w

∂r∂t

∂δu

∂t +∂u

∂t

∂²δw

∂r∂t + 1

r

∂²w

∂θ ∂t

∂δυ

∂t +1 r

∂υ

∂t

∂²δw

∂θ ∂t

+I2

∂²w

∂r∂t

∂²δw

∂r∂t + 1 r²

∂²w

∂θ ∂t

∂²δw

∂θ ∂t

dA (16) Now, the following governing equations will be derived by use of equations (13) and (16) in Hamilton’s

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principle

∂Nrr

∂r +1 r

∂Nrθ

∂θ +Nrr−Nθ θ

r −I0

∂²u

∂t² +I1

∂³w

∂r∂t² =0 (17)

∂Nrθ

∂r +1 r

∂Nθ θ

∂θ +2Nrθ

r −I0

∂²υ

∂t² +I1

1 r

∂³w

∂θ ∂t² =0 (18)

∂Qr

∂r +1 r

∂Qθ

∂θ +Qr

r −I0

∂²w

∂t² =0 (19)

∂Mrr

∂r +1 r

∂Mrθ

∂θ +Mrr−Mθ θ

r

−Q_r+I₂ ∂³w

∂r∂t² −I₁∂²u

∂t² =0 (20)

∂M_rθ

∂r −1 r

∂Mθ θ

∂θ −2M_rθ r +Qθ

−I2

r

∂³w

∂θ ∂t² +I1

∂²υ

∂t² =0 (21)

QrandQθare transverse shear intensities and can be obtained from equations (20) and (21). Replacing the strain components in the resultant force and moments give

Nrr=Q⁰₁₁∂u

∂r +Q⁰₁₂ u

r +1 r

∂υ

∂θ

−Q¹₁₁∂²w

∂r²

−Q¹₁₂ 1

r

∂w

∂r + 1 r²

∂²w

∂θ²

(22) Nθ θ=Q⁰₁₂∂u

∂r +Q⁰₂₂ u

r +1 r

∂υ

∂θ

−Q¹₁₂∂²w

∂r²

−Q¹₂₂ 1

r

∂w

∂r + 1 r²

∂²w

∂θ²

(23) Nrθ =Q₃₃⁰

∂υ

∂r +1 r

∂u

∂θ −υ r

+Q¹₃₃ 2

r²

∂w

∂θ − 2 r²

∂²w

∂r∂θ

(24) M_rr=Q₁₁¹ ∂u

∂r +Q₁₂¹ u

r +1 r

∂υ

∂θ

−Q₁₁² ∂²w

∂r²

−Q²₁₂ 1

r

∂w

∂r + 1 r²

∂²w

∂θ²

(25) Mθ θ=Q₁₂¹ ∂u

∂r +Q₂₂¹ u

r +1 r

∂υ

∂θ

−Q₁₂² ∂²w

∂r²

−Q²₂₂ 1

r

∂w

∂r + 1 r²

∂²w

∂θ²

(26) M_rθ =Q¹₃₃

∂υ

∂r +1 r

∂u

∂θ −υ r

+Q²₃₃ 2

r²

∂w

∂θ − 2 r²

∂²w

∂r∂θ

(27)

where the coefficients are defined as Q^k_ij=

h/2

−h/2z^kCijdz, i,j=1, 2, 3, k=0, 1, 2 (28) Substituting the above equations into the equilib- rium equations (17) to (21) and neglecting the longi- tudinal inertia terms leads to the following governing equations of motion

Q⁰₁₁ ∂²u

∂r² − u r² − 1

r²

∂υ

∂θ +1 r

∂²υ

∂r∂θ +1 r

∂u

∂r

+Q⁰₃₃

−1 r

∂²υ

∂r∂θ + 1 r²

∂²u

∂θ² − 1 r²

∂υ

∂θ

+Q¹₁₁

−∂³w

∂r³ + 2 r³

∂²w

∂θ² − 1 r²

∂³w

∂r∂θ² + 1

r²

∂w

∂r −1 r

∂²w

∂r²

=0 (29)

Q⁰₁₁ 1

r

∂²u

∂r∂θ + 1 r²

∂²υ

∂θ² + 1 r²

∂u

∂θ

+Q¹₁₁

−1 r

∂³w

∂r²∂θ − 1 r³

∂³w

∂θ³ − 1 r²

∂²w

∂r∂θ

+Q⁰₃₃

−1 r

∂²u

∂r∂θ − υ r² + 1

r²

∂u

∂θ +∂²υ

∂r² +1 r

∂υ

∂r

=0 (30) Q²₁₁∇_r²∇_r²w+Q¹₁₁

u r³ − 1

r²

∂u

∂r +∂³u

∂r³ + 1 r³

∂³υ

∂θ³

+ 1 r³

∂²u

∂θ² +2 r

∂²u

∂r² + 1 r³

∂υ

∂θ − 1 r²

∂²υ

∂r∂θ + 1

r

∂³υ

∂r²∂θ + 1 r²

∂³u

∂r∂θ²

=I0

∂²w

∂t² (31)

where ∇_r² is the Laplace operator in the polar co- ordinate system. In the above equations of motion, the in-plane and transverse deformations are coupled together. By the following non-dimensional parame- ters

R= r

a, = θ

α, U =u

a, V = υ

a, W = w a (32) the governing differential equations of motion can be rewritten as

Q₁₁⁰ a

∂²U

∂R² − U R² − 1

R²α

∂V

∂+ 1 Rα

∂²V

∂R∂ +1 R

∂U

∂R

+Q⁰₃₃ a

− 1 Rα

∂²V

∂R∂+ 1 R²α²

∂²U

∂² − 1 R²α

∂V

∂

+Q¹₁₁ a²

−∂³W

∂R³ + 2 R³α²

∂²W

∂² − 1 R²α²

∂³W

∂R∂² + 1

R²

∂W

∂R − 1 R

∂²W

∂R²

=0 (33)

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Q₁₁⁰ a

1 Rα

∂²U

∂R∂+ 1 R²α²

∂²V

∂² + 1 R²α

∂U

∂

+Q₁₁¹ a²

− 1 Rα

∂³W

∂R²∂ − 1 R³α³

∂³W

∂³ − 1 R²α

∂²W

∂R∂

+Q₃₃⁰ a

− 1 Rα

∂²U

∂R∂− V R² + 1

R²α

∂U

∂

+ ∂²V

∂R² + 1 R

∂V

∂R

=0 (34)

Q₁₁²

a³ ∇_R²∇_R²W +Q₁₁¹ a²

U R³ − 1

R²

∂U

∂R +∂³U

∂R³ + 1

R³α³

∂³V

∂³ + 1 R³α²

∂²U

∂² + 2 R

∂²U

∂R² + 1 R³α

∂V

∂

− 1 R²α

∂²V

∂R∂+ 1 Rα

∂³V

∂R²∂+ 1 R²α²

∂³U

∂R∂²

=I₀a∂²W

∂t² (35)

where∇_R²is the non-dimensional Laplace operator.

3 BOUNDARY CONDITIONS

It should be noticed that beside the above coupled PDEs, there are four BCs at each side of the plate. These BCs are any proper combinations of the geometric or the natural BCs. In this article, the cases that contain the combinations of clamped, simply supported, and free are studied. The BCs on the radial edges can be written as

Clamped

U =0, V =0, W =0, W=0 (36) Simply supported

U =0, V =0, W =0, Mθ θ=0 (37) Free

Nrθ =0, Nθ θ=0, Q2+∂Mrθ

∂r =0, Mθ θ=0 (38) and on the circular edges are

Clamped

U =0, V =0, W =0, WR =0 (39) Simply supported

U =0, V =0, W =0, Mrr=0 (40) Free

Nrr =0, Nrθ =0, Q1+1 r

∂Mrθ

∂θ =0, Mrr =0 (41)

Now, the synchronous motion in which the general shape of the plate does not change with time is considered. Mathematically, this implies that the unknown functions U, V, and W are separable in space and time, namely

U(R,,t)=U(R,)(t) (42) V(R,,t)=V(R,)(t) (43) W(R,,t)=W(R,)(t) (44) Substituting the above relations into equations (33) to (35) leads to

_tt+ω²=0 (45)

which shows a harmonic motion with the frequencyω and the following coupled ODEs

Q₁₁⁰ a

∂²U

∂R² − U R² − 1

R²α

∂V

∂+ 1 Rα

∂²V

∂R∂ +1 R

∂U

∂R

+Q⁰₃₃ a

− 1 Rα

∂²V

∂R∂+ 1 R²α²

∂²U

∂² − 1 R²α

∂V

∂

+Q¹₁₁ a²

−∂³W

∂R³ + 2 R³α²

∂²W

∂² − 1 R²α²

∂³W

∂R∂² + 1

R²

∂W

∂R − 1 R

∂²W

∂R²

=0 (46)

Q₁₁⁰ a

1 Rα

∂²U

∂R∂ + 1 R²α²

∂²V

∂² + 1 R²α

∂U

∂

+Q¹₁₁ a²

− 1 Rα

∂³W

∂R²∂− 1 R³α³

∂³W

∂³ − 1 R²α

∂²W

∂R∂

+Q⁰₃₃ a

− 1 Rα

∂²U

∂R∂− V R² + 1

R²α

∂U

∂

+ ∂²V

∂R² + 1 R

∂V

∂R

=0 (47)

Q₁₁²

a³ ∇_R²∇_R²W+Q¹₁₁ a²

U R³ − 1

R²

∂U

∂R +∂³U

∂R³ + 1

R³α³

∂³V

∂³ + 1 R³α²

∂²U

∂² + 2 R

∂²U

∂R² + 1 R³α

∂V

∂

− 1 R²α

∂²V

∂R∂+ 1 Rα

∂³V

∂R²∂+ 1 R²α²

∂³U

∂R∂²

= −I0aω²W (48)

In order to find the solution, these equations and corresponding BCs should be discretized.

4 DQ ANALOGUE

According to DQ analogue, the sector plate is discretized into a set of grid pointsNR andNinr and θdirections, respectively. In this method, each partial derivative with respect to an independent variable is

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expressed as [24]

∂^mf(R,)

∂R^m =

NR

n=1

A^R(m)_in f(Rn,_j), i=1,. . .,NR,

j=1,. . .,N (49)

∂^mf(R,)

∂^m =

N

n=1

A^(m)_jn f(Ri,_n), i=1,. . .,NR,

j=1,. . .,N (50)

where A^x(m)_pq are the weighting coefficients related to the mth order derivative. The off-diagonal and the diagonal weighting elements related to the first-order derivative are defined as follows

A_pq^x(1)= M(xp)

(xp−xq)M(xq) forp=q (51) A_pp^x(1)= −

Nx

q=1q=p

A^x(1)_pq forp=q, p,q=1, 2,. . .,Nx

(52) wherexstands for the independent variable(R,)that the partial derivative is defined with respect to it. Also, M(x_p)is defined as

M(xp)=

Nx

q=1 q=p

(xp−xq) (53)

The following relationship is given for evaluating the weighting coefficients of higher-order derivatives

A_pq^x(m)=m

A_pp^(m−1)A⁽¹⁾_pq − A_pq^(m−1) xp−xq

forp=q (54)

A_pp^x(m)= −

Nx

q=pq=1

A_pq^x(m) forp=q, p,q=1, 2,. . .,Nx

(55) In the present study, the Chebyshev–Gauss–Lobatto distribution is assumed to locate the grid points as

Ri= 1 a

b+1

2

1−cos

(i−1)π

NR−1 (a−b)

,

i=1, 2,. . .,NR (56)

_j= 1 2

1−cos

(j−1)π

N−1 , j=1, 2,. . .,N (57) There are three degrees of freedom at each grid point, while the number of BCs is four. Therefore, for implementation of the BCs, following the procedure given in references [32] and [33], these extra degrees of freedom on boundaries of the plate are introduced For circumferential edges

K¹= ∂²W

∂R²

For radial edges K² =∂²W

∂² (58)

The equations of motion can be discretized as follows

Q₁₁⁰ a

!_N R n=1

A^R(2)_in Unj−Uij

R_i² − 1 R²_iα

N

n=1

A⁽¹⁾_jn Vin

+ 1 Riα

N

k=1 NR

n=1

A^R(1)_in A_jk⁽¹⁾Vnk+ 1 Ri

NR

n=1

A_in^R(1)Unj

"

+Q⁰₃₃ a

!

− 1 Riα

N

k=1 NR

n=1

A^R(1)_in A⁽¹⁾_jk Vnk

+ 1 R²_iα²

N

n=1

A_jn⁽²⁾Uin− 1 R_i²α

N

n=1

A_jn⁽¹⁾Vin

"

+Q¹₁₁ a²

!

−A_i1^R(1)K_1j¹−

NR

k=1 NR−1

n=2

A_in^R(1)A^R(2)_nk Wkj

−A^R(1)_iN_RK_N¹_R_j+ 2 R³_iα²

N

n=1

A_jn⁽²⁾Win

− 1 R²_iα²

N

k=1 NR

n=1

A_in^R(1)A⁽²⁾_jk Wnk

+ 1 R²_i

NR

n=1

A_in^R(1)W_nj− 1 Ri

NR

k=1

A^R(2)_in W_nj

"

=0 (59)

Q₁₁⁰ a

! 1 Riα

N

k=1 NR

n=1

A^R(1)_in A⁽¹⁾_jk Unk

+ 1 R²_iα²

N

n=1

A_jn⁽²⁾Vin+ 1 R_i²α

N

n=1

A⁽¹⁾_jn Uin

"

+Q⁰₃₃ a

!

− 1 R_iα

N

k=1 NR

n=1

A^R(1)_in A⁽¹⁾_jk Unk−Vij

R_i² + 1

R²_iα

N

n=1

A⁽¹⁾_jn Uin+

NR

n=1

A_in^R(2)Vnj

+ 1 Ri

NR

n=1

A_in^R(1)Vnj

"

+Q¹₁₁ a²

− 1

R_i³α³A⁽¹⁾_j1 K_i1²

− 1 R³_iα³

N

k=1 N−1

n=2

A⁽¹⁾_jn A⁽²⁾_nk Wik

− 1

R³_iα³A_jN⁽¹⁾_T K_iN²_T − 1 Riα

N

k=1 NR

n=1

A_in^R(2)A⁽¹⁾_jk Wnk

− 1 R²_iα

N

k=1 NR

n=1

A^R(1)_in A_jk⁽¹⁾Wnk

"

=0 (60)

(7)

Q₁₁² a³

!

A_i1^R(2)K_1j¹ +

NR

k=1 NR−1

n=2

A^R(2)_in A_nk^R(2)Wkj

+A^R(2)_iN_RK_N¹_R_j+ 1 R³_i

NR

n=1

A_in^R(1)Wnj

− 1 R²_i

NR

k=1

A_in^R(2)Wnj+ 2 Ri

A^R(1)_i1 K_1j¹

+ 2 Ri

NR

k=1 NR−1

n=2

A^R(1)_in A^R(2)_nk Wkj+ 2 Ri

A^R(1)_iN_RK_N¹_R_j

+ 4 R⁴_iα²

N

n=1

A_jn⁽²⁾Win− 2 R³_iα²

N

k=1 NR

n=1

A^R(1)_in A_jk⁽²⁾Wnk

+ 2 R²_iα²

N

k=1 NR

n=1

A^R(2)_in A_jk⁽²⁾Wnk

+ 1

R⁴_iα⁴A_j1⁽²⁾K_i1²+ 1 R⁴_iα⁴

Table 1 Physical properties of metal and ceramic constituents in FGM [34]

Young’s Mass Poisson’s

modulus density ratio

Constituent E(GPa) ρ(kg/m³) ν

Al2O3 393.0 3970.0 0.25

Ni 199.5 8900.0 0.3

×

N−1 n=2

N

k=1

A⁽²⁾_jn A⁽²⁾_nk Wik+ 1

R⁴_iα⁴A_jN⁽²⁾K_iN²

"

+Q¹₁₁ a²

!Uij

R_i³ − 1 R²_i

NR

n=1

A_in^R(1)Unj+

NR

n=1

A^R(3)_in Unj

+ 1 R³_iα³

N

n=1

A⁽³⁾_jn Vin+ 1 R_i³α²

N

n=1

A⁽²⁾_jn Uin

+ 2 Ri

NR

n=1

A_in^R(2)Unj+ 1 R_i³α

N

n=1

A⁽¹⁾_jn Vin

− 1 R²_iα

N

k=1 NR

n=1

A_in^R(1)A⁽¹⁾_jk Vnk

+ 1 R_iα

N

k=1 NR

n=1

A_in^R(2)A⁽¹⁾_jk Vnk

+ 1 R²_iα²

N

k=1 NR

n=1

A^R(1)_in A_jk⁽²⁾Unk

"

= −I0aω²Wij (61)

5 DISCRETIZED FORM OF BOUNDARY CONDITIONS

By the same procedure, the BCs can also be discretized.

The BCs on the radial edges are written as fori=2,. . .,NR−1, j=1 and N

Table 2 Comparison of frequency parameterωa²#

ρh/Dfor homogeneous, fully clamped circular and annular sectorial plates [35]

Mode number Source of

ν α b/a result 1 2 3 4 5

0.3 π/2 0.000 01 Present 48.7856 87.7788 104.8845 136.9268 164.5724

Reference [35] 48.786 87.779 104.89 136.93 164.58

0.3 π/2 0.5 Present 95.2195 114.9385 150.5208 201.3455 253.1905

Reference [35] 95.220 114.94 150.52 201.36 253.20

0.33 π/3 0.000 01 Present 75.6323 145.1486 148.8054 234.0209 243.4977

Reference [35] 75.63 145.6 148.8 237.2 244.1

0.33 π/3 0.2 Present 75.6929 146.3871 148.8054 240.9279 243.5847

Reference [35] 75.69 146.4 148.8 241.8 243.9

0.33 π/3 0.4 Present 85.2498 150.0954 194.2151 243.5874 266.0402

Reference [35] 85.25 150.1 194.2 243.9 266.0

0.33 π/3 0.6 Present 152.0316 192.9298 265.7419 368.6179 398.7130

Reference [35] 152.03 192.93 265.75 368.64 398.71

Table 3 Comparison of frequency parameterωa²#

ρh/Dfor homogeneous annular sectorial plates with radial edges simple supported,ν=0.3,α=π/4, andb/a=0.5 [35]

Mode number Circumferential Source of

edges results 1 2 3 4 5 6

Free Present 21.0672 66.7220 81.6035 146.4127 176.1176 176.9023

Reference [35] 21.067 66.722 81.604 146.41 176.12 176.90

Simple supported Present 68.3792 150.9822 189.5986 278.3857 283.5926 387.6154

Reference [35] 68.379 150.98 189.60 278.39 283.59 387.64

Clamped Present 107.5670 178.8170 269.4914 305.8442 346.4612 476.3033

Reference [35] 107.58 178.82 269.49 305.84 346.46 476.30