Free Vibration Analysis of Functionally Graded Moderately Thick Annular Sector Plates Using Differential Quadrature Method
S.H. Mirtalaie
1,a, M.A. Hajabasi
2,band F. Hejripour
3,c1 Department of mechanical engineering, Islamic Azad University, Najaf Abad Branch, Najaf Abad, Isfahan, Iran
2,3 Department of mechanical engineering, Shahid Bahonar University of Kerman, Kerman, Iran e-mail: a[email protected], [email protected], c[email protected]
Keywords: Free vibration; Functionally Graded Material; Moderately Thick Annular Sector Plate;
Differential Quadrature Method
Abstract. In this paper, the free vibration of moderately thick annular sector plates made of functionally graded materials is studied using the Differential Quadrature Method (DQM). The material properties of the functionally graded plate are assumed to vary continuously through the thickness, according to a power-law distribution. The governing differential equations of motion are derived based on the First order Shear Deformation plate Theory (FSDT) and then solved numerically using DQM under different boundary conditions. The results for the isotropic plates which are derivable with this approach are presented and compared with the literature and they are in good agreement. The natural frequencies of the functionally graded moderately thick annular sector plates under various combinations of clamped, simple supported and free boundary conditions are presented for the first time. The effects of boundary conditions, sector angle, radius ratio, thickness to outer radius ratio, volume fraction exponent and variation of the Poisson’s ratio on the free vibration behavior of the plate are studied
Introduction
In recent years, the new microscopic inhomogeneous composites which 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). The continuity in the material properties of these new types of composites provides better mechanical behavior in comparison with the fiber-reinforced composites. The mismatch of material properties across the interface of two discrete materials in fiber-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 FGM. Moreover a combination of the properties of the metal and ceramic can be achieved by the composition of them. These properties that consist of high-temperature resistance due to low thermal conductivity, wear and oxidation resistance for ceramics and the high toughness, high strength, machinability and bonding capability for metals, cause that FGMs can resist high-temperature conditions while their toughness maintains.
Because of these good characteristics, FGMs have extensively used in various industries such as space structures, turbo-machinery, nuclear and chemical industries, defence mechanisms, energy conversion systems, tribology [1], systems in vibration and acoustic controls or condition monitoring [2] and semiconductor devices [3]. Due to this widespread applicability, FGMs have been extensively studied by researchers in recent years, particularly the bending and vibration analyses of functionally graded structures like as plates are carried out by many researchers. As known by authors, the vibration problem of annular sector-shaped plates made of Functionally Graded materials are less regarded so far whereas components with sectorial geometries are extensively used in engineering structures such as aeronautical and naval structures, nuclear reactors, curved bridge decks and panels. Therefore the free vibration analysis of sector plates is of practical importance in the structural mechanics and there are many papers in the literature that
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 31.57.61.141-20/11/11,17:10:37)
have been devoted to this topic. Recently sector and annular sector plates made of FGM are studied by a few researchers; Jomehzadeh et al. [4] presented an exact analytical approach for bending analysis of functionally graded annular sector plates based on the first order shear deformation plate theory. Sahraee [5] carried out the bending analysis of functionally graded thick circular sector plates based on the Levinson plate theory and the first order shear deformation plate theory. Nie and Zhong [6] studied the three-dimensional free and forced vibration of functionally graded annular sectorial plates with simply supported radial edges and arbitrary circular edges using a semi- analytical approach. Recently Hosseini-Hashemi et al. [7] investigated the buckling and free vibration behaviors of radially functionally graded circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. To the best of author’s knowledge the problem of vibration of functionally graded moderately thick annular sector plates has not been studied until now. In this paper the Differential Quadrature Method (DQM) is used for free vibration analysis of functionally graded moderately thick annular sector plates. The DQM which was introduced by Bellman and Casti [8] is a powerful method for solving initial and boundary value problems that needs less computational efforts comparing with the other numerical methods such as finite element method and finite difference method. The method firstly used by Bert et al [9] 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.
Governing Equations
The differential equations of motion for free vibrations of the FG moderately thick sector plates can be easily derived using Hamilton’s principle. The Young’s modulus of elasticity E(z) , Poisson’s ratio υ(z) and the density ρ(z) of the Functionally Graded Material is considered to vary through the thickness of the plate according to the following polynomial distributions
kL m
c
m h
L z L L z
L )
2 )(1 (
)
( = + − + (1) where L stands for E,υ and ρ . Also, the subscripts c and m indicate the properties of the ceramic and metal constituents, respectively. Also h is the thickness of the plate in the z direction and k is named the power-law or volume fraction exponent of the FGM. In the cylindrical coordinate system, based on the first-order shear deformation plate theory for the moderately thick annular sector plate, the displacement field is assumed to take the following form
) , , ( ) , , ( ) , , ,
~(r z t u r t z r t
u θ = θ + ψr θ (2)
) , , ( ) , , ( ) , , ,
~(r zt v r t z r t
v θ = θ + ψθ θ (3)
) , , ( ) , , ,
~(r z t wr t
w θ = θ (4) whereu(r,θ,t), v(r,θ,t) and w(r,θ,t) are the radial, circumferential displacements and transverse deflection of the point (r,θ) on the mid-plane (z=0) of the plate at the timet. Also, ψr and ψθ are rotation functions about θ and r-axis, respectively. The geometry of the plate, dimensions and the coordinate system which are used are shown in the Fig. 1. Based on the above displacement components the linear strain components of the elastic material are defined as
r r r
rr u, zψ ,
ε = + , εrz=w,r+ψr, εθθ ,θ + + (ψ +ψθ,θ)
= r
r z r
u v
θ θ
θ ψ
ε = +
r w
z
, , ( )
, ,
, ,
r r
r
r v z r
r v u
θ θ θ θ
θ ψ ψ ψ
ε − +
+
− +
= (5) where the subscripts after commas denote differentiation with respect to the independent variables. Considering the material properties Eq.1, the two dimensional stress-strain law for plane stress condition can be written as
2
1 1 2 2 3 3 4 4 5 5 1 2 2 1
{ } ( ) [ ] { }
1 ( )
1 ( )
1, , ( )
2 0
T T
r r r r z z r r r r z z
i j
E z E
z
E E E E E z E E z
O t h e r E
θ θ θ θ θ θ θ θ
σ σ σ σ σ ε ε ε ε ε
ν
υ υ
= −
= = = = = − = =
=
(6)
In order to use the Hamilton’s principle, first the following resultant in-plane forces Nij, bending moments Mij and shear forces Q, are considered as
∫
−= /2
2 / (1, ) )
,
( h
h ij
ij
ij M z dz
N σ , Q Q h dz
h rz z
r =
∫
−/22
/ ( , )
) ,
( θ κ σ σθ (7)
Figure 1. Geometry of the plate and coordinate system
where i and j stand for r and θ respectively. Also κ is the shear correction factor and is assumed to be π2/12. Substituting the strain components from the Eq. (5) in Hamilton’s principle, using integration by parts and simplifying the results, leads to the following governing equations of motion
2 2 2 2
11 2 2 2 33 2 2 2
2 2
2 2
14 2 2 2 36 2 2 2
2 2
1 2 2 2
1 1 1 1 1 1
1 1 1 1 1 1
r r r r
r
u u v u v u v v
C C
r r r r r r r r r r r
C C
r r r r r r r r r r r
I u I
t t
θ θ θ θ
θ θ θ θ θ
ψ ψ ψ ψ
ψ ψ ψ ψ
θ θ θ θ θ
ψ
∂ + ∂ − ∂ − + ∂ + ∂ − ∂ − ∂ +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ + ∂ − ∂ − + ∂ + ∂ − ∂ − ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂
= ∂ +
∂ ∂
(8)
2 2 2 2
11 2 2 2 33 2 2 2
2 2 2 2
14 2 2 2 36 2 2 2
2 2
1 2 2 2
1 1 1 1 1 1
1 1 r 1 r 1 r 1 r 1
v u u u u v v v
C C
r r r r r r r r r r r
C C
r r r r r r r r r r r
I v I
t t
θ θ θ θ
θ
θ θ θ θ θ
ψ ψ ψ ψ ψ ψ ψ ψ
θ θ θ θ θ
ψ
∂ + ∂ + ∂ + ∂ − ∂ +∂ − + ∂ +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + − + − +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂
= ∂ +
∂ ∂
(9)
2 2 2 2
14 2 2 2 36 2 2 2
2 2
2 2
44 2 2 2 66 2 2 2
2 2
33 2 2 3
1 1 1 1 1 1
1 1 1 1 1 1
( )
r r r r
r
u u v u v u v v
C C
r r r r r r r r r r r
C C
r r r r r r r r r r r
w u
C I I
r t
θ θ θ θ
θ θ θ θ θ
ψ ψ ψ ψ
ψ ψ ψ ψ
θ θ θ θ θ
κ ψ ψ
∂ + ∂ − ∂ − + ∂ + ∂ − ∂ − ∂ +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ + ∂ − ∂ − + ∂ + ∂ − ∂ − ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂
∂ ∂
− + = +
∂ ∂ 2
r
∂t
(10)
2 2 2 2
14 2 2 2 36 2 2 2
2 2 2 2
44 2 2 2 66 2 2 2
2
33 2 2 3
1 1 1 1 1 1
1 1 1 1 1 1
( 1 )
r r r r
v u u u u v v v
C C
r r r r r r r r r r r
C C
r r r r r r r r r r r
w v
C I I
r t
θ θ θ θ
θ
θ θ θ θ θ
ψ ψ ψ ψ ψ ψ ψ ψ
θ θ θ θ θ
κ ψ
θ
∂ + ∂ + ∂ + ∂ − ∂ +∂ − + ∂ +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ + ∂ + ∂ + ∂ − ∂ +∂ − + ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂
∂ ∂
− + = +
∂ ∂
2
t2
ψθ
∂
(11)
2 2 2
33 2 2 2 1 2
1 1 1
r r w w w w
C I
r r r r r r r t
ψθ
ψ ψ
κ θ θ
∂ ∂ ∂ ∂ ∂ ∂
+ + + + + =
∂ ∂ ∂ ∂ ∂ ∂
(12)
In which the inertia terms has the following form
. 3 , 2 , 1 , ) (
2 /
2 /
1 =
=
∫
−− z dz i
z I h
h i
i ρ (13) and the coefficients Cij are defined as
/ 2
/ 2
11(11),12 (12),33(33), 0 , ( ) 14 (11),15(12),36 (33), 1 44 (11), 45(12), 66 (33), 2
h k
ij mn
h
k
C z E dz ij mn k
− k
=
= = =
=
∫
(14) where Eij are components of the coefficient matrix in the stress strain relation.Boundary conditions
It should be noticed that beside the above coupled PDEs, there are five boundary conditions at each boundary of the plate. These boundary conditions are any proper-combinations of the geometric or the natural BCs. In this paper, the cases that contain the combinations of clamped, simple supported and free are studied. The boundary conditions on the radial edges of the plate are given as
clamped u=0, v=0, w=0, ψr=0, ψθ=0 (15)
simple
supported u=0, v=0, w=0, ψr=0, Mθ =0 (16)
free Nθ =0, Nrθ =0,Mθ =0, Mrθ =0, Qθ=0 (17)
And the boundary conditions on the curved edge are given as
clamped u=0, v=0, w=0, ψr=0, ψθ =0 (18)
simple
supported u=0, v=0, w=0, Mr=0, ψθ =0 (19)
free Nr =0, Nrθ =0,Mr =0, Mrθ =0, Qr=0 (20)
In the following using the DQ-process the equations of motion under any arbitrary boundary conditions are discreted and the solution procedure are explained.
DQ-Analogue
According to DQ-analogue, the sector plate is discretized into a set of Nrand Nθ grid points inr, θ directions, respectively. In this method, each partial derivative with respect to a variable is expressed as follows [9]
( )
θ θ
θ A f r i N j N
r r f
r j
n N
n m m in
m R
,..., 1 , ,..., 1 , ) , , (
1 )
( = =
∂ =
∂
∑
=
(21) ( )
θ θ
θ
θ A f r i N j N
r f
r n
i N
n m m jn
m
,..., 1 , ,..., 1 , ) , , (
1 )
( = =
∂ =
∂
∑
Θ=
(22) where A(mpq)are the weighting coefficients related to the mthorder derivative. The off-diagonal weighting elements and the diagonal weighting elements related to the first order derivative are defined as follows, respectively
q p x for M x x
x A M
q q p
p
pq ≠
= −
) ( ) (
)
) (
1
( (23)
x N
p q q
pq
pp A for p q pq N
A , , 1,2,...,
1 ) 1 ( )
1
( =−
∑
= =≠=
(24)
where x stands for the independent variable (r,θ)that the partial derivative is defined with respect to it. M(xp) is defined as
∏
≠=
−
= N
p q q
q p
p x x
x M
1
) ( )
( (25) The following relationship is given for evaluating the weighting coefficients of higher order derivatives
q p x for x A A A m A
q p
m pq pq m pp m
pq ≠
− −
=
−
−
) 1 ( ) 1 ( ) 1 ( )
( (26)
x N
p q q
m pq
pp A for p q p q N
A , , 1,2,...,
1 ) ( )
1
( =−
∑
= =≠=
(27) In the present study the Gauss-Chebyshev-Lobatto is used to locate the grid points
r r
i a b i N
N b i
r )]( ), 1,2,...,
1 ) 1 cos(( 1 2[
1 − =
−
− − +
= π
(28)
θ θ
θ π j N
N j
j )] , 1,2,...,
1 ) 1 cos(( 1 2[
1 =
−
− −
= (29) Considering the synchronous motion for the free vibrations of the plate and following the DQ- procedure the equations of motion can be discretized and here only the discrete form of the first and last equation is presented for the sake of brevity
( 2 ) (1) (1) (1) (1)
1 1 2 2
1 1 1 1 1
( 2 ) (1) (1) (1)
3 3 2 2
1 1 1 1
( 2 ) 1 4
1
1 1 1
( )
1 1 1
( )
( 1
R R R
R
R
N N
N N N
ij
im m j im m j jm im im jn m n
m i m i m i i m n
N N N N
jm im jm im im jn m n
m m m n
i i i
N
im r m j i
m i
C A u A u A v u A A v
r r r r
C A u A v A A v
r r r
C A A
ψ r
Θ Θ
Θ Θ Θ
= = = = =
= = = =
=
+ − − +
+ − −
+ +
∑ ∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
∑
(1) 2 (1) 2 (1) (1)1 1 1 1
( 2 ) (1) (1) (1) 2
3 6 2 2 1 2
1 1 1 1
1 1
)
1 1 1
( ) ( )
R R
R
ij
N N
N N
r ij
m r m j jm im im jn m n
m i m i i m n
N N N N
jm r im jm im im jn m n ij r
m m m n
i i i
A A A
r r r
C A A A A I u I
r r r
θ θ
θ θ
ψ ψ ψ ψ
ψ ψ ψ ω ψ
Θ Θ
Θ Θ Θ
= = = =
= = = =
− − +
+ − − = − +
∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
(30)
(1) (1) ( 2 ) (1)
33
1 1 1 1
( 2 ) 2
2 1 1
1 1
(
1 )
R N R R
N N N
r ij
im r mj jm im im mj im mj
m i i m m i m
N
jm im ij
i m
C A A A w A w
r r r
A w I w
r
θ
κ ψ ψ ψ
ω
Θ
Θ
= = = =
=
+ + + +
+ = −
∑ ∑ ∑ ∑
∑
(31)
where ωis the frequency of the harmonic motion.
Discretized form of Boundary conditions
Following the procedure described above, the boundary conditions can be discretized too.
Because of considering the briefness, only the discretized forms of the clamped and simple supported boundary conditions are presented here. The boundary conditions on the radial edges are written as
: 1
, 1 , ,
2 − = Θ
= N j andN
i
for … R
Clamped uij=0, vij=0, wij =0, =0
rij
ψ , =0
θij
ψ (32)
Simple supported
=0
uij , vij=0, wij =0, =0
rij
ψ ,
(33)
0 )]
1( [
)]
1( [
1 ) 1 ( 1
) 1 ( 44
1 ) 1 ( 66 1
) 1 ( 36
1 ) 1 ( 1
) 1 ( 14
= + +
+
−
−
+ +
=
∑
∑
∑
∑
∑
∑
Θ Θ
=
=
=
=
=
=
ij im R
mj R
mj R
R
r N
m jm i N
m r im
N
m r im N
m mj im
ij N
m im jm i N
m mj im
r A A
C
A C u A C
u v r A
u A C M
ψ ψ ψ
ψ
θ θ
The boundary conditions on the curved edges can also be descritized in a similar manner.
Solution of the System of Equations
To create the eigenvalue system of equations, the discretized form of the equations of motion and boundary conditions can be rearranged by separating the degrees of freedom into the boundary and the domain degrees of freedom. The governing equations of motion in discrete form can be rewritten as
d d
b A Y M Y
Y
A]{ } [ ]{ } [ ]{ }
[ 1 + 2 =ω2 (34) where the subscripts d and b denotes the values at domain and boundary grid points, respectively. Also {Y}is the vector of degrees of freedom which considered as
[
u v w r]
TY}= ψ ψθ
{ (35) and [M] is the matrix of inertia terms. Also the boundary conditions can be rearranged as
0 } ]{
[ } ]{
[A3 Y b+ A4 Y d= (36) Evaluating the vector {Y}b from the BCs and substituting the result in governing EQs leads to the following eigenvalue system of equations
d
d M Y
Y A A A
A] [ ][ ] [ ]){ } [ ]{ }
([ 2 − 1 4 −1 3 =ω2 (37) Solution of this eigenvalue system of equations represents the natural frequencies and mode shapes of the plate under consideration.
Numerical Results
For the metal and ceramic constituents, Ni and Al2O3 are selected and their mechanical properties are summarized in Table I [10].
Considering any proper combinations of the natural and essential boundary conditions and following the procedure described in the previous section, the natural frequencies and mode shapes of the considered plate are obtained. To validate and confirm the accuracy of the solution procedure, the numerical results are calculated for a special case of FG-plate by setting
=0
=
=kρ kν
kE , which coincides with the isotropic plate. These results are tabulated in Table II for various boundary conditions and as it can be seen they show a good agreement. Also the convergence of solutions is shown in Table III for the plates with various boundary conditions. This table truly illustrates the effectiveness of the method. It can be seen that the numerical results have been rapidly converged in the most cases. The non-dimensional natural frequencies Ω=ωa2 I0/Dc
of the FG-plate are presented in Table IV for various BC’s, aspect ratios, thickness to outer radius ratios and sector angles. The parameter Dc =Ech3/[12(1−νc2)] is the flexural rigidity of the plate corresponding to the ceramic constituent. The letters C, S and F are abbreviations for the clamped, simple supported and free boundary conditions and the order of these letters are in CCW direction on the boundaries which start from the inner radius of the plate. As the inner radius of the sector plate tends to zero, the annular sector plate coincide with the solid sector plate which its results are presented in first row of each state by setting b/a=1×10−5. Focusing on the data in this table indicates that the increase/decrease of the natural frequencies is compatible with the nature of BCs.
The influence of the variation of the Poisson’s ratio on the natural frequencies of the plate is studied. The results are tabulated in Table V. As it can be seen, the variation of the Poisson’s ratio according to the polynomial distribution has a little effect on the frequency parameters. In the majority of papers devoted to the problem of the vibration of the FG-plates, the Poisson’s ratio is assumed to be constant. The results presented in Table V confirm that this assumption is reasonable.
Finally, the effect of the variation of the volume fraction exponent on the free vibration behavior of the plate under various boundary conditions is shown in Table VI. It can be seen that the increase of the volume fraction exponent leads to decrease of the frequency parameter of the plate.
TABLE I. PHYSICAL PROPERTIES OF THE CONSTITUENTS IN FGM
Constituent E(GPa) ρ(kg/m3) ν
3 2O
Al 393.0 3970.0 0.25
Ni 199.5 8900.0 0.3
TABLE II. COMPARISON OF FREQUENCY PARAMETER ωa2 ρh/D FOR ISOTROPIC ANNULAR SECTORIAL PLATES WITH VARIOUS BOUNDARY CONDITIONS [11]
Bc’s α b/a h/a Source
of result Mode Number
1 2 3 4
CCCC π/6 0.25 0.01 present 187.0559297.3371412.5457424.3504 Ref. [11] 187.056 297.337 412.547 424.348
3 /
2π 0.5 0.2 present 49.7875 54.6741 63.2535 73.0471 Ref. [11] 49.788 54.674 63.253 74.838
SSSS π/6 0.1 0.01 present 97.8169 183.3218276.6866286.8812 Ref. [11] 97.817 183.322 276.687 286.881
3
π/ 0.25 0.1 present 38.1220 84.4475 87.0090 139.8201 Ref. [11] 38.122 84.447 87.009 139.820
CSCS π/3 0.1 0.1 present 45.8402 91.7642 93.4502 144.2694 Ref. [11] 45.840 91.764 93.450 144.269
3 /
2π 0.5 0.2 present 49.2957 53.0988 60.8130 72.1520 Ref. [11] 49.296 53.099 60.813 72.152
CFCF π/6 0.5 0.1 present 69.3399 83.5299 157.0073159.2101 Ref. [11] 69.340 83.530 157.007 159.210
3
π/ 0.1 0.2 present 19.1104 25.5578 43.7895 53.8179 Ref. [11] 19.111 25.558 43.790 54.719
TABLE III. CONVERGENCE OF THE FREQUENCY PARAMETER Ω FOR FG ANNULAR SECTORIAL PLATE WITH VARIOUS BOUNDARY CONDITIONS, b/a=0.5,h/a=0.1,α=π/3,kν =kE=kρ =1.
Bc’s Number of Nodes
Mode Sequence Number
1 2 3 4
CCCC
14 71.3895 101.3669 144.2230 147.7890 15 71.3895 101.3669 144.2231 147.7890 16 71.3895 101.3670 144.2231 147.7890 17 71.3895 101.3670 144.2231 147.7890
SSSS
12 44.4110 76.6235 119.5728 121.8487 13 44.4110 76.6234 119.5727 121.8533 14 44.4110 76.6235 119.5727 121.8527 15 44.4110 76.6235 119.5727 121.8525 16 44.4111 76.6235 119.5727 121.8526 17 44.4111 76.6235 119.5727 121.8526
FSFS
22 9.7409 36.6014 37.2460 71.8984 23 9.7405 36.6017 37.2460 71.8974 24 9.7409 36.6016 37.2462 71.8977 25 9.7408 36.6018 37.2459 71.8975 26 9.7408 36.6018 37.2459 71.8975
TABLE IV. FREQUENCY PARAMETER Ω FOR FG CIRCULAR AND ANNULAR SECTORIAL PLATES WITH VARIOUS BOUNADRY CONDITIONS.
β b/a h/a Mode Number
1 2 3 4
Fully clamped
6
π/ 0.00001 0.01 0.2 160.7367 71.5614 255.5345 95.9597 354.5860 105.3043 364.6833 114.5408
0.5 0.01 164.0761 291.0560 354.6856 483.6465 0.1 114.6001 179.5028 204.6127 234.6718
3
π/ 0.00001 0.01 0.1 64.8159 54.0905 124.1779 94.1086 127.2991 96.1891 199.8160 138.6566
0.2 0.1 54.1702 95.1163 96.1896 142.6668 0.2 39.8989 64.0346 64.4817 75.7061
Fully simple supported
3
π/ 0.00001 0.01 0.1 34.6598 32.4896 81.3328 70.7066 84.2705 72.9430 144.8615 116.0432
0.2 0.01 34.9661 83.8815 84.2719 152.4631 0.2 28.1954 55.7187 56.0203 75.6137
3 /
2π 0.00001 0.01 0.2 16.5281 14.9962 34.4145 27.7681 51.1500 37.9118 56.9636 41.7257
0.5 0.1 35.5393 44.2808 58.2759 76.5574
0.2 30.1772 36.6224 46.4189 58.4378
FSFS
6
π/ 0.2 0.01 0.1 41.6013 38.0767 105.7530 87.6859 151.8003 118.5004 181.9077 138.1487
0.5 0.01 41.2670 99.6485 151.6961 173.8616 0.2 31.8239 59.9917 71.0179 81.9797
3
π/ 0.2 0.1 0.2 10.6123 9.9533 37.5145 31.3698 41.1217 33.7039 75.0150 45.1577
0.5 0.01 10.0348 40.7223 40.9075 88.2145 0.1 9.7400 36.6022 37.2460 71.8976
SCSC
6
π/ 0.00001 0.01 0.1 141.6629 102.4051 232.5796 152.3612 327.8204 196.6934 337.6629 203.1491
0.5 0.1 102.4060 161.6064 196.7131 234.5898 0.2 67.3998 102.7331 113.9824 114.9379
3
π/ 0.00001 0.01 0.1 52.7622 45.7108 108.0541 85.7398 110.9904 87.5816 179.4453 130.6766
0.2 0.01 52.7536 107.9444 110.9904 179.7579 0.2 35.0414 60.4356 60.9957 75.6391
TABLE V. THE EFFECT OF THE VARIATION OF THE POISSON’S RATIO ON THE FREQUENCY PARAMETER FOR FG ANNULAR SECTORIAL PLATE WITH VARIOUS BOUNDARY CONDITIONS,
3 / , 005 . 0 / , 5 . 0
/a= h a= α =π
b .
Bc’s
ModeNumber 5.0=n 1=n 2=n
kν=n kν=0 kν=n kν=0 kν=n kν=0
CCCC
1 95.4521 94.9498 91.0923 90.4214 87.8793 87.0922 2 143.3229 142.5694 136.7759 135.7695 131.9502 130.7698 3 221.0422 219.8817 210.9437 209.3938 203.4983 201.6807 4 237.0666 235.8230 226.2353 224.5745 218.2480 216.3005
SSSS
1 50.6591 50.4237 48.6039 48.2988 46.9999 46.6388 2 93.2821 92.8077 89.2323 88.6104 86.1755 85.4456 3 158.4579 157.6519 151.3166 150.2450 146.0183 144.7619 4 161.2113 160.3763 154.0693 152.9665 148.7312 147.4370
SCSC
1 60.8287 60.5212 58.1841 57.7796 56.1896 55.7139 2 117.8393 117.2166 112.5406 111.7146 108.6080 107.6393 3 164.3747 163.5337 156.9425 155.8232 151.4372 150.1248 4 197.4403 196.3983 188.5075 187.1223 181.8946 180.2701
FSFS
1 10.4703 10.4834 10.0350 10.0517 9.6909 9.7105 2 42.8436 42.8799 40.7945 40.8417 39.2865 39.3411 3 42.9833 42.9798 40.9722 40.9684 39.4830 39.4768 4 92.8216 92.8262 88.4625 88.4680 85.2339 85.2398