an accurate mathematical study on the free vibration of stepped thickness circular
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An Accurate Mathematical Study on the Free Vibration of Stepped Thickness CircularTRANSCRIPT
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Accepted Manuscript
An accurate mathematical study on the free vibration of stepped thickness circular/annular Mindlin functionally graded plates
Shahrokh Hosseini-Hashemi, Masoud Derakhshani, Mohammad Fadaee
PII: S0307-904X(12)00458-1DOI: http://dx.doi.org/10.1016/j.apm.2012.08.002Reference: APM 9035
To appear in: Appl. Math. Modelling
Received Date: 30 July 2011Revised Date: 25 July 2012Accepted Date: 15 August 2012
Please cite this article as: S. Hosseini-Hashemi, M. Derakhshani, M. Fadaee, An accurate mathematical study onthe free vibration of stepped thickness circular/annular Mindlin functionally graded plates, Appl. Math.Modelling (2012), doi: http://dx.doi.org/10.1016/j.apm.2012.08.002
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An accurate mathematical study on the free vibration of
stepped thickness circular/annular Mindlin functionally
graded plates
Shahrokh Hosseini-Hashemi
Impact Research Laboratory, School of Mechanical Engineering,, Iran University of
Science and Technology, Narmak, Tehran 16848-13114, Iran.
Masoud Derakhshani
Impact Research Laboratory, School of Mechanical Engineering,, Iran University of
Science and Technology, Narmak, Tehran 16848-13114, Iran.
Mohammad Fadaee
Impact Research Laboratory, School of Mechanical Engineering,, Iran University of
Science and Technology, Narmak, Tehran 16848-13114, Iran.
Total Number of Pages: 30
Total Number of Figures: 5
Total Number of Tables: 7
Corresponding Author: Mohammad Fadaee
Postal Address
Impact Research Laboratory, School of Mechanical Engineering,, Iran University of
Science and Technology, Narmak, Tehran 16848-13114, Iran.
E-mail address: [email protected]
Tel.: +98 9131701907; fax: +98 2177 240 488
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An accurate study on the free vibration of stepped thickness
circular/annular Mindlin functionally graded plates
Abstract
An analytical solution based on a new exact closed form procedure is presented
for free vibration analysis of stepped circular and annular FG plates via first order shear
deformation plate theory of Mindlin. The material properties change continuously
through the thickness of the plate, which can vary according to a power-law distribution
of the volume fraction of the constituents, whereas Poissons ratio is set to be constant.
Based on the domain decomposition technique, five highly coupled governing partial
differential equations of motion for freely vibrating FG plates were exactly solved by
introducing the new potential functions as well as using the method of separation of
variables. Several comparison studies were presented by those reported in the literature
and the FEM analysis, for various thickness values and combinations of stepped thickness
variations of circular/annular FG plates to demonstrate highly stability and accuracy of
present exact procedure. The effect of the geometrical and material plate parameters such
as step thickness ratios, step locations and the power law index on the natural frequencies
of FG plates is investigated.
Keywords:
Free vibration; Stepped circular/annular plate; Functionally graded material; Mindlin theory
1. Introduction
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Because of the potential savings on material usage, weight reduction of the plates
and an increase in the stiffness, circular and annular plates of variable thickness are of
practical interest in many fields of engineering, including civil, mechanical, and
aerospace engineering. These plates are often subjected to dynamic loads that cause
vibrations and the analysis of them has difficulty related to the variable thickness. Hence,
it is very important to have an accurate procedure for the free vibration analysis of
circular and annular plates with variable thickness.
A wide range of research has been carried out on free vibration of circular and annular
plates with variable thickness which mostly used the classical plate theory (i.e., CPT) and
a numerical solution method. A systematic summary of research studies on free vibration
of thin circular and annular plates with variable thickness, through the 1960s until the
mid-1980s, are made of Leissa [1-4]. Singh and Saxena [5] employed the RayleighRitz
method to investigate free flexural vibration of a quarter of a circular thin plate with
variable thickness. Singh and Jain [6] also presented free asymmetric transverse
vibrations of a non-uniform polar orthotropic annular sectorial plate on the basis of the
CPT. A new version of the DQM was extended by Wang and Wang [7] to analyze the
free vibration of thin circular sectorial plates with six combinations of boundary
conditions. Wang [8] presented the generalized power series solution for the vibration
analysis of classical circular plates with variable thickness. A finite element analysis of
the lateral vibration of thin annular and circular plates with variable thickness is
developed by Chen and Ren [9]. Wu et al. [10] and Wu and Liu [11] studied the free
vibration of stepped and variable thicknesses solid circular plates, respectively, using the
generalized differential quadrature rule (GDQR). Al-Jumaily and Jameel [12] determined
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the natural frequencies of simply supported and clamped stepped-thickness plates using
classical plate solutions with exact continuity conditions at the step. Liang et al. [13]
extended a new method using the limited finite element method (FEM) for the analysis of
the natural frequencies of circular/annular plates of polar orthotropy, stepped and variable
thickness.
In order to eliminate the deficiency of the CPT for moderately thick and thick circular
and annular plates, especially to the case of the plate with variable thickness, the first
shear deformation theory (FSDT), the third order shear deformation theory (TSDT), the
higher order shear deformation theory (HSDT) and the 3D elasticity solution including
the effects of transverse shear deformation and rotary inertia, were employed by many
research groups using analytical and numerical methods. Xiang and Zhang [14] presented
the exact solutions for vibration of circular Mindlin plates with multiple step-wise
thickness variations. Duan et al. [15] modified the fundamental mode of the stepped
circular plate from a twisting mode shape to an axisymmetric mode shape by adjusting
the rigidity of the edge annular plate. They applied the Mindlin plate theory to describe
the dynamic behavior of the stepped circular plate. Hang et al. [16] presented exact
vibration frequencies, mode shapes, and modal stress resultants of vibrating, stepped
circular plates with free edges. Hosseini-Hashemi et al. [17] proposed the exact closed-
form frequency equations and transverse displacement for thick circular plates with free,
soft simply supported, hard simply supported and clamped boundary conditions based on
Reddys third-order shear deformation theory. Kang and Leissa [18] and Kang [19] used
Ritz method for 3D analyses of linearly and nonlinearly thickness variation of annular
plates, respectively.
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In recent years, functionally graded materials (FGMs) have gained such popularity as
special composites with material properties that vary continuously through their
thickness. Typically, FGMs are made of a ceramic and a metal. The gradual change of the
material properties results in eliminating discontinuities of stresses, high resistance to
temperature gradients, reduction in residual and thermal stresses, high wear resistance,
and an increase in strength to weight ratio.
According to the aforementioned FG properties, circular and annular plates made of
FGMs are of great interest for researchers and designers, recently. Hosseini-Hashemi et
al. [20] employed the differential quadrature method (DQM) to analyze free vibration
analysis of radially functionally graded circular and annular sectorial thin plates of
variable thickness, resting on the Pasternak elastic foundation. Also, Hosseini-Hashemi et
al. [21] developed an exact closed-form frequency equation for free vibration analysis of
circular and annular moderately thick FG plates with constant thickness based on the
Mindlins first-order shear deformation plate theory. Gupta et al. [22] presented free
vibration analysis of non-homogeneous circular plates of variable thickness using the
FSDT. Chebyshev collocation technique has been employed to obtain the natural
frequencies and mode shapes. Efraim and Eisenberger [23] analyzed free vibration of
variable thickness thick annular isotropic and FGM plates using the exact element
method. Tajeddini et al. [24] described three-dimensional free vibration behavior of thick
circular and annular isotropic and functionally graded (FG) plates with variable thickness,
resting on Pasternak foundation using the polynomial-Ritz method. Based on the first-
order shear deformation theory, Tornabene [25] and Tornabene et al. [26] focused on the
dynamic behavior of moderately thick functionally graded conical, cylindrical shells and
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annular plates. The discretization of the system equations by means of the Generalized
Differential Quadrature (GDQ) method is used to solve a standard linear eigenvalue
problem.
The beneficial literature review reveals that the study of circular and annular FG plates
with stepped and/or variable thickness is very limited in number, especially to the case of
the analytical solutions for moderately thick plates. This observation may be due to the
fact that in FG plates, unlike isotropic plates, the stretching and bending equations are
highly coupled and the obtaining of an analytical solution becomes more complicated.
Thus, it is important to understand the exact dynamic behavior of circular and annular FG
plates of stepped and/or variable thickness. Also, to the best of the authors' knowledge,
there is no literature for exact closed-form solutions of vibration analysis of stepped
thickness circular/annular Mindlin FG plates. Therefore, the authors attempt to fill this
apparent void.
This paper presents an exact closed form solution for free vibration of stepped thickness
moderately thick circular and annular FG plates. Introducing the new auxiliary and
potential functions as well as using a new exact analytical approach [21], the equations of
motion are exactly solved without any usage of approximate methods. The merit and the
high accuracy of the current exact approach are validated by comparing the results of the
present FSDT with the available data in literature and a finite element analysis (FEA) for
circular and annular Mindlin plates with different boundary conditions and various
combinations of step configurations. Finally, the effect of the plate parameters such as
step thickness ratios, step locations and the power law index on the natural frequencies of
FG plates is considered.
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2. Mathematical formulation
2.1 Geometrical configuration and material properties
Consider an annular functionally graded plate of radius nr and thickness nh consists of n
steps in which the radius and thickness of ith annular segment is ir and ih respectively,
as shown in Fig. 1. The plate geometry and dimensions are defined in an orthogonal
cylindrical coordinate system ( , , )r z to extract the mathematical formulations.
The FGMs are composite materials, the mechanical properties of which vary gradually
due to changing the volume fraction of the constituent materials usually in thickness
direction. In this study, the properties of the plate are assumed to vary through the plate
thickness with a power-law distribution of the volume fractions of two materials. Unless
mentioned otherwise, the top surface of the first segment ( 1 / 2z h= ), which assumed
here the thickest part of the plate, is metal-rich whereas the bottom surface of the same
segment ( 1 / 2z h= ) is ceramic-rich. By considering this assumption, Young's modulus
and mass density are assumed vary through the plate thickness as
( ) ( ) ( )m c f cE z E E V z E= +
(1a)
( ) ( ) ( )m c f cz V z = +
(1b)
where
1
1( ) ( )
2
g
f
zV z
h= +
(2)
in which the subscripts m and c represent the metallic and ceramic constituents,
respectively. fV shows the volume fraction and g is the power-law index which takes
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only non-negative values. For 0g = and g = , the plate is fully ceramic and metallic,
respectively, whereas the composition of metal and ceramic is linear for 1g = . Poissons
ratio is taken as 0.3 throughout the analysis.
2.2 Displacement field
A stepped annular Mindlin plate with n steps can be divided to n annular plates. Let us
consider an annular Mindlin plate of radius ir and thickness ih which refers to the ith
segment of the stepped annular FG plate as depicted in Fig. 1. According to the FSDT, in
which the in-plane displacements of the plate are expanded as linear functions of the
thickness coordinate and the transverse deflection is constant through the plate thickness,
the displacement field is used for the ith segment of the stepped annular Mindlin FG plate
as follows
0( , , , ) ( , , ) ( , , )i i i
ru r z t u r t z r t = +
(3a)
0( , , , ) ( , , ) ( , , )i i iv r z t v r t z r t = +
(3b)
0( , , , ) ( , , ) ( , , )i i iw r z t w r t w r t = =
(3c)
where i-overscript indicates the ith segment of stepped circular/annular plate, ,i iu v and
iw denote the displacements in r , and z directions, respectively. 0iu and 0iv denote the
in-plane displacements of mid-plane in radial and circumferential directions. Also, ir
and i show the slope rotations in r-z and -z planes at z=0 for ith segment of the plate,
respectively. Based on FSDT assumptions, the strain and stress in z direction are
neglected. Hence, iw is used instead of 0iw throughout the analysis.
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2.3 Equations of motion
The exact vibration of circular/annular Mindlin FG plate has been studied by the first
author [21], recently. In this study, the same analytical approach is used to solve the free
vibration of stepped circular/annular moderately thick FG plates. The equations of motion
of the ith annular segment of the stepped circular/annular plate can be written as [21]
i
r
iiiii
r
i
r
i
r IuIr
NN
r
N
r
N
201 +=
+
+
(4a)
iiiii
r
i
r
i
IvIr
N
r
N
r
N
2012 +=+
+
(4b)
iii
r
ii
r wIr
Q
r
Q
r
Q 1=+
+
(4c)
i
r
iiii
r
ii
r
i
r
i
r IuIQr
MM
r
M
r
M
302 +=
+
+
(4d)
iiiiii
r
i
r
i
IvIQr
M
r
M
r
M
3022 +=+
+
(4e)
where
2
2
1 2 3
2
( , , ) ( )(1, , )i
i
h
i i i
h
I I I z z z dz
= (5a)2
2
( , , ) ( , , )i
i
h
i i i i i i
r r rr r
h
N N N dz
= (5b)2
2
( , , ) ( , , )i
i
h
i i i i i i
r r rr r
h
M M M zdz
= (5c)2
2
( , ) ( , )i
i
h
i i i i
r rz z
h
Q Q dz
= (5d)
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in which ,i ik kN M and i
kQ (k=r, , r) denote the stress resultants and ik (k=r, , r)
show the normal and shear stresses. Also, ikI (k=1,2,3) denote the inertias and dot-
overscript indicates differentiation with respect to the time.
2.4 Equations of motion in dimensionless form
For generality and convenience in deriving mathematical formulations, the following
non-dimensional terms are defined
, , , ,i i ii i i i
n i n n n
r z h h rR Z
r h r h r = = = = =
(6a-e)
For harmonic motion, the displacement fields in dimensionless forms are taken as
( , , , )( , , )
ii j t
i
i
u r z tu R Z e
h
= (7a)
( , , , )( , , )
ii j t
i
i
v r z tv R Z e
h
= (7b)
( , , )( , )
ii j t
n
w r tw R e
r
= (7c)
and also we have
tjiitji
r
i
r
tj
i
iitj
i
ii eee
h
vve
h
uu
==== ,,, 0000(8a-d)
where
0( , , )i i i
i ru R Z u Z = + (9a)
0( , , )i i i
iv R Z v Z = + (9b)
( , )i iw R w = (9c)
Introducing stress resultants in dimensionless forms as
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, , ,i
i j tk
k
c i
NN e k r r
E h
= = (10a)
2, , ,
i
i j tk
k
c i
MM e k r r
E h
= = (10c)
, ,i
i j tk
k
c i
QQ e k r
E h
= = (10c)
By substituting the stress-strain relations in polar coordinate [17] into Eqs. (10a)-(10c),
the following equations are obtained
0 0 0
1 2 2 3( , ) ( , )( ( )) ( , )( ( ))
i i i i i i
i i i i i i r r
r r i
u u vN M K K K K
R R R R R R
= + + + + +
(11a)
0 0 0
1 2 2 3( , ) ( , ) ( ) ( , ) ( )
i i i i i i
i i i i i i r r
i
u v uN M K K K K
R R R R R R
= + + + + +
(11b)
0 0 0
1 2 2 3
1( , ) ( , )( ) ( , )( )
2
i i i i i i
i i i i i i r
r r i
v u vN M K K K K
R R R R R R
= + + +
(11c)
2
1
1( )
2
i
i i i
r r
wQ K
R
= +
(11d)
2
1
1( )
2
i
i i i wQ KR
= +
(11e)
where
2
2
1 2 3 2
2
( )( , , ) (1, , )
1
i
i
h
i i i
h
E zK K K z z dz
=
(12a)
, 1, 2,3i
i kk k
c i
KK k
E h= =
(12b)
By substituting Eqs. (11a)-(11e) into Eqs. (4a)-(4e), the final form of equations of motion
are obtained as follows
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2 2 2
0 0 0 0 0 0 0 01 2 2 2 2 2 2
2 22 2
2 2 2 2 2 2 2 2
1
2
1
2
i i i i i i i ii
i i i ii i i ii r r r r
u u u v v u v vK
R R R R R R R R R R R
KR R R R R R R R R R R
S
+ + +
+ + + + =
2
1 1 0 2( )i i i i i
i rI u I +
(13a)
2 2 2 2
0 0 0 0 0 0 0 01 2 2 2 2 2 2
2 22 2
2 2 2 2 2 2 2
2
1 1
1
2
1
2
(
i i i i i i i ii
i i i ii i i ii r r r r
i i
i
u u v u u v v vK
R R R R R R R R R R R
KR R R R R R R R R R R
S I v
+ + + + + +
+ + + + + + + = 0 2 )
i i iI +
(13b)
2 2 2
0 0 0 0 0 0 0 02 2 2 2 2 2 2
2 22 2
3 2 2 2 2 2 2 2
2
1
2
1
2
i i i i i i i ii
i i i ii i i ii r r r r
u u u v v u v vK
R R R R R R R R R R R
KR R R R R R R R R R R
S
+ + +
+ + + +
2
1 2 0 3( ) ( )i
i i i i i i i
r i r
wS I u I
R + = +
(13c)
2 2 2 2
0 0 0 0 0 0 0 02 2 2 2 2 2 2
2 22 2
3 2 2 2 2 2 2
2
1
2
1
2
(
i i i i i i i ii
i i i ii i i ii r r r r
ii i
u u v u u v v vK
R R R R R R R R R R R
KR R R R R R R R R R R
wS
R
+ + + + + +
+ + + + + + +
+ 21 2 0 3) ( )
i i i i i
iS I v I = +
(13d)
2 2
2 1 1
ii ii i i i ir r
i iS w S I wR R R
+ + + =
(13e)
where
( ) ( ) ( )1/2 21 2 31/2
, , 1, ,i i i i i ic
zI I I Z Z dZ
= (14)2 2
2 2 2R R R R = + +
(15)
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( )( )
( )2 32
1 2
1 2 22 2
1, , ,
212 1 12 1
i
i i c i c iii n i
i i
K h E hS S r D
D
= = = =
(16a-d)
2.5 Exact solution for the transverse displacement iw
In order to solve the five highly coupled differential equations of motion (i.e. Eqs. (13a)-
(13e)), the following steps must be taken to uncouple Eqs. (13a)-(13e)
1. Eq. (13a) is differentiated with respect to R.
2. Eq. (13a) is divided by R.
3. Eq. (13b) is first differentiated with respect to and then divided by R. 4. Two auxiliary functions
1 2and i i are defined as
0 0 01 2,
i i i ii ii i r r
u u v
R R R R R R
= + + = + +
(17a-b)
5. If three equations obtained from steps (1) to (3) are added together, we will obtain
2
1 1 2 2 1 1 1 2 2( )i i i i i i i i i
iK K S I I + = + (18)
6. Doing the above five steps on Eqs. (13c) and (13d), respectively, yields
2
2 1 3 2 2 2 1 2 1 3 2( ) ( )i i i i i i i i i i i i
iK K S w S I I + + = + (19)
7. Eq. (13e) must be rewritten by using Eqs. (17b) as
( )2 22 2 1 1i i i i i ii iS w S I w + = (20)8. The next step in the analysis is to determine the functions
1
i and 2i using Eqs. (18)
and (19).
9. Using the equations obtained from steps (5) to (8) and after some mathematical
manipulation, a sixth-order partial differential equation with constant coefficients is
acquired in terms of iw as follows
-
1 2 3 4 0i i i i i i i iA w A w A w A w + + + = (21)
where the coefficients ikA (k=1,2,3,4) are determined by
( )21 2 21 3 2i i i i iiA S K K K= (22a)( ) ( )2 2 22 1 1 1 3 2 2 1 3 3 1 2 22i i i i i i i i i i i i ii iA S I K K K S K I K I K I = + + (22b)
( ) ( )( )2 2 2 23 1 1 1 1 3 3 1 2 2 2 1 3 2 2 1 12i i i i i i i i i i i i i i i i ii i iA S S I K I K I K I S I I I S K I = + + (22c) ( )2 4 2 24 1 1 1 1 3 2 2 1[ ]i i i i i i i i ii iA S I S I I I S I = (22d)
The function ( ),iw R can be written as
( ) ( ) ( ), cosi iw R w R p = (23)in which the non-negative integer p represents the circumferential wave number of the
corresponding mode shape. By substituting Eq. (23) into Eq. (21), the reduced following
form of the equation is obtained as
( )( )( )1 2 3 ( ) 0i i i ix x x w R = (24)where
2 2
2 2 p
R R R R
= +
1 2,i ix x and 3
ix are the roots of the following equation
3 2
1 2 3 4 0i i i iA x A x A x A+ + + = (25)
The general solution of Eq. (21) can be expressed as the summation of three Bessel
functions as follows
( ) ( ) ( )1 1 2 2 3 3 0 , 0 , 0i i i i i ix w x w x w = = = (26a-c)1 2 3( )
i i i iw w w wR + += (27)
in which
-
( ) ( ) ( )1 1 2 2 3 3 cos , cos , cosi i i i i iw w p w w p w w p = = = (28)In this study, Cardanos formula [27] is used to solve the obtained third-order Eq. (25):
At the first, the following transformation is introduced to eliminate the second-order term
in Eq. (25)
2
13
i
i
Ax y
A=
(29)
Substituting this transformation, the new form of Eq. (25) is obtained as follows
3 0i iy a y b+ + = (30)
setting
1 1
3 31 1,
2 2
i i i i i ib d b d = + = (31a-b)
where id is positive square root of
3 21 1
3 2
i i id a b
= + (32)
The Cardano formulas for the roots 1 2,y y and 3y are
21
13
ii i i
i
Ax
A = + (33a)
( ) ( )221
1 13
2 3 2
ii i i i i
i
Ax j
A = + + (33b)
( ) ( )231
1 13
2 3 2
ii i i i i
i
Ax j
A = + (33c)
It can be observed that practically for all range of the frequency, the Eq. (25) has three
real solutions.
Finally, the solution to Eq. (21) can be written as
-
( ) ( )31
1 3 2, [ , ( , )]cos( )i i i i i i i
k k k k k k
k
w R c w p R c w p R p +=
= + (34)
where
i i
k kx = (35a)
( )( )1
, 01,2,3
, 0
i i
p k ki
ki i
p k k
J R xw k
I R x
(35b)
( )( )2
, 01, 2,3
, 0
i i
p k ki
ki i
p k k
Y R xw k
K R x
(35c)
i
kc are unknown coefficients and pJ and pY are the Bessel functions of the first and
second kind, respectively, whereas pI and pK are the modified Bessel functions of the
first and second kind, respectively.
2.6 Exact solutions for 0 0, ,i i i
ru v and i
Four auxiliary functions 1 2 3, ,i i if f f and 4
if are introduced as follows
1 1 0 2
i i i i i
rf K u k = + (36a)
2 1 0 2
i i i i if K v K = + (36b)
3 2 0 3
i i i i i
rf K u k = + (36c)
4 2 0 3
i i i i if K v K = + (36d)
By using Eqs. (36a)-(36d) in Eqs. (13a)-(13e), the equations of motion can be rewritten
as follows
2 2 2
1 1 1 2 2 1 2 2
2 2 2 2 2 2
1 1 2 3
1
2
i i i i i i i i
i i i i
f f f f f f f f
R R R R R R R R R R R
G f G f
+ + + = +
(37a)
-
2 2 2 2
1 1 2 1 1 2 2 2
2 2 2 2 2 2
1 2 2 4
1
2
i i i i i i i i
i i i i
f f f f f f f f
R R R R R R R R R R R
G f G f
+ + + + + + = +
(37b)
2 22
3 3 3 34 4 4 4
2 2 2 2 2 2
3 1 4 3 5
1
2
i i i ii i i i
ii i i i i
f f f ff f f f
R R R R R R R R R R R
wG f G f G
R
+ + +
= + +
(37c)
2 22 2
3 3 3 34 4 4 4
2 2 2 2 2 2
3 2 4 4 5
1
2
i i i ii i i i
ii i i i i
f f f ff f f f
R R R R R R R R R R R
wG f G f G
R
+ + + + + +
= + +
(37d)
23 31 1 2 46 7 5 8
i ii i i ii i i i i i
i
f ff f f fG G G w G w
R R R R R R
+ + + + + = + (37e)
where
( )21 3 1 2 21 2
2 1 3
i i i i i
ii
i i i
S K I K IG
K K K
=
( )21 1 2 2 1
2 2
2 1 3
i i i i i
ii
i i i
S K I K IG
K K K
=
( )21 3 2 2 3 2 23 2
2 1 3
i i i i i i i
ii
i i i
S K I K I S KG
K K K
+=
( )21 1 3 2 2 2 1
4 2
2 1 3
i i i i i i i
ii
i i i
S K I K I S KG
K K K
=
5 2
i iG S= 22 2
6
2 1 3
i ii i
i i i
S KG
K K K
=
2
2 17
2 1 3
i ii i
i i i
S KG
K K K
=
2
8 1 1
i i i
iG S I=
(38a-h)
In order to determine 1 2 3, ,i i if f f and 4
if , the following forms of solution are considered
3 51 2 41 1 2 3 4 5
i ii i ii i i i i iw ww w wf a a a a a
R R R R R
= + + + +
(39a)
3 51 2 42 1 2 3 4 5
i ii i ii i i i i iw ww w wf b b b b b
R R R R R
= + + + +
(39b)
3 51 2 43 6 7 8 9 10
i ii i ii i i i i iw ww w wf a a a a a
R R R R R
= + + + +
(39c)
-
3 51 2 44 6 7 8 9 10
i ii i ii i i i i iw ww w wf b b b b b
R R R R R
= + + + +
(39d)
in which ika and
i
kb are unknown coefficients. Also, 4iw and 5
iw are unknown functions.
By substituting Eqs. (39a)-(39d) into Eqs. (37a)-(37e), the coefficients ika and
i
kb as well
as the functions 4
iw and 5iw can be determined as follows
( )2 5
2
1 4 1 4 2 3
5 1 5
2
, 1, 2,3
( ), 6,7,8
i i
i i i i i i i i
k ki i
k ki i i
k k
i
G Gk
x G G x G G G Ga b
x G ak
G
=
+ + = =
=
(40a-b)
( ) sin , 4,5i ik kw w p k= = (41)where
( ) ( )4 7 41 4 8 42 4 , ,i i i i i i iw c w p R c w p R = + (42a)( ) ( )5 9 51 5 10 52 5 , ,i i i i i i iw c w p R c w p R = + (42b)
and
, 4,5i ik kx k = = (43a)
2 2
1 1 2 1 1 2
4 5
4 4,
2 2
i i i i i i
i ix x +
= =
(43b)
( )( )
1 4 1 4 2 31 2 2
2 4( ),
1 1
i i i i i ii i
G G G G G G
+
= =
(43c)
( )( )1
, 0, 4,5
, 0
i i
p k ki
ki i
p k k
J R xw k
I R x
(43d)
( )( )2
, 0, 4,5
, 0
i i
p k ki
ki i
p k k
Y R xw k
K R x
(43e)
-
2
1
, 4,51
( )2
1 , 9,10
i
i i i i
k k k
Gk
a b x G
k
=
= = =
(44)
Finally, the exact solutions for 0 0, ,i i i
ru v and i according to Mindlin's theory, are
obtained as follows
2 3 3 10 2
2 1 3
i i i ii
i i i
K f K fu
K K K
=
(45a)
2 4 3 20 2
2 1 3
i i i ii
i i i
K f K fv
K K K
=
(45b)
2 1 1 3
2
2 1 3
i i i ii
r i i i
K f K f
K K K =
(45c)
2 2 1 4
2
2 1 3
i i i ii
i i i
K f K f
K K K
=
(45d)
2-7 Satisfaction of the continuity and boundary conditions
To apply the continuity conditions, the dimensional forms of the displacement
components and stress resultants must be used instead of the dimensionless ones. This is
because the annular segments of the stepped plate have different values of the thicknesses
ih . Based on the FSDT, ten continuity conditions should be satisfy for each step location,
which can be written as follows
1i iw w += 10 0
i iu u +=
1
0 0
i iv v += 1i ir r +=
1i i
+= 1i ir rQ Q +=
1i i
r rN N+
= 1i iN N +
=
-
1i i
r rM M+
= 1i iM M +
=
(46a-j)
Both inner and outer edges of the stepped circular/annular plate can take any
combinations of classical boundary conditions, including free, soft simply supported,
hard simply supported and clamped. Based on the FSDT, the classical boundary
conditions can be written as follows
Free edge
0, 0, 0 , 0 , 0i i i i ir r rQ N N M M = = = = = (47a)
Soft simply supported edge
0, 0, 0 , 0 , 0i i i i ir rw N N M M = = = = = (47b)
Hard simply supported edge
00, 0, 0 , 0 , 0i i i i i
r rw v N M= = = = = (47c)
Clamped edge
0 00, 0, 0, 0 , 0i i i i i
rw u v = = = = = (47d)
It should be noted that, for stepped circular FG plates, second type of Bessel function
becomes singular at r=0. Hence, the unknown coefficients of them (4 5 6 8 10, , , ,i i i i ic c c c c )
should be equal to zero. Therefore, the iw can be written as follows
( ) ( ) ( )11
3
, [ , cos )]i i i ik k kk
w R c w p R p =
= (48)
( ) ( )4 7 41, sini i iw R c w p = (49a)
( )5 8 51, sin( )i i iw R c w p = (49b)
-
Substituting Eqs. (45a)-(45d) into four appropriate boundary conditions (i.e., Eqs.(47a)-
(47d)) along the edges 0
r r= and n
r r= as well as the satisfaction of the continuity
conditions lead to a coefficient matrix. For a nontrivial solution, the determinant of the
coefficient matrix must be set to zero for each p. Solving the eigenvalue equations yields
the frequency parameters .
3. Numerical results
This section contains two parts; firstly, the authors try to validate the present
solution with aid of the different examples including various step thickness ratios, step
locations, the power law index and different boundary conditions in Section 3.1. After
verification of results, the effects of the geometrical and material properties of stepped
thickness circular/annular plates on the frequency parameters will be discussed in Section
3.2. For convenience of notation, an annular plate is described by symbolism defining the
boundary conditions at their edges, for example, SC indicates an annular plate which is
restricted by hard simply support and clamped boundary conditions in the inner and outer
edges, respectively. S and Ss are as symbols of hard and soft simply support, respectively.
It should be noted that in this paper Poissons ratio is assumed to be 0.3. The numbers in
parentheses (m,n) show that the vibrating mode has m nodal diameters and vibrates in the
nth mode for the given m value. Two types of FG plates are used in this study which their
material properties are listed in Table. 1. In all FSDT comparison results, the shear
correction factor has been taken to be 5/6. Unless mentioned otherwise, all natural
frequencies of stepped thickness FG plates are considered to be dimensionless as
nncnDhr /2 = . Also, a well-known commercially available FEM package was
used for the extraction of the frequency parameters.
-
3.1. Comparison results
In this section, according to a beneficial literature review and using a reliable FE
model, the comparison studies are provided to validate the results of the present study
and demonstrate its accuracy.
3.1.1 FG annular plate
In order to investigate the efficiency of the present exact solution as well as the
stability of the computer code, the results are compared with those obtained by Sh.
Hosseini-Hashemi et al. [21] and the 3D finite element results for FG annular plate
without any step variation as shown in Table 2. The comparison study is performed for
first ten natural frequencies (Hz) of C-C FGM1 annular plate ( )(1,)(2.010
mrmr == )
when 2.0,1.0,01.0=h . Results show that, for all the natural frequencies, the present exact
results are identical to those reported by Hosseini-Hashemi et al. [21]. This is due to the
fact that the procedure of the present solution and those of Hosseini-Hashemi et al. [21] is
exactly the same. Also, it can be deduced from Table 2 that the present exact results are
very close to the highly accurate numerical 3D finite element solution even if the plate is
thick.
3.1.2 Stepped isotropic circular plate
Tables 3 and 4 show the comparison study of first ten frequency parameters for
one and two step variations circular plates with those obtained by Xiang and Zhang [14],
respectively. Numerical results have been performed for isotropic circular plates with
-
clamped, hard simply supported and free boundary conditions. In Table 3, for each case
of the boundary conditions, the step thickness ratios 21 / hh are taken to be 0.5, 1.5 and 2.
The step location ratio 21 / rr is fixed at 5.0 and the plate thickness ratio 22 / rh is set to be
0.1. In Table 4, the plate thickness ratio 33 / rh is set to be 0.1 and the step location ratios
are fixed at 3/1/ 31 =rr and 3/2/ 32 =rr . From Tables 3 and 4, it is evident that there is
an excellent agreement among the results confirming the high accuracy of the current
analytical approach. As it is seen, in all cases, the discrepancy between the present exact
results and those reported by Xiang and Zhang [14] is equal to zero. It stems from the fact
that, in both methods, the procedure of solution is exact. Because of existing coupling
between the stretching and bending in stepped thickness circular/annular Mindlin
functionally graded plates, the free vibration analysis of plate is much more complicated
than that of the isotropic plate. It is worth of noting that all the results listed in Tables 3
and 4 are the out of plane modes of plate. It means that the exact method of Xiang and
Zhang [14] gives only the out of plane modes of stepped isotropic plates while the
present exact procedure provided the in-plane and out of plane modes of stepped
isotropic and FG circular/annular plates. This will be discussed in Section 3.1.3.
3.1.3 Stepped FG circular/annular plates
A comparative study for evaluation of first ten natural frequencies (Hz) of stepped FGM1
circular /annular plates between the present exact solution and the finite element analysis
is carried out in Tables 5-7. In Tables 5 and 6, numerical results have been calculated for
free, hard simply supported and clamped FGM1 circular Mindlin plates with one and two
step variations, respectively while Table 7 shows the natural frequencies of F-C and S-S
-
FGM1 annular plates with one step variation. The power law index g is equal to 1. The
step locations and thicknesses are selected as follows
For FGM1 circular plate with one step variation (Table 5):
)(2,)(1.0,)(1,)(2.0 2211 mrmhmrmh ==== .
For FGM1 circular plate with two step variations (Table 6):
)(4,)(16.0,)(2,)(2.0,)(1,)(24.0 332211 mrmhmrmhmrmh ======
For FGM1 annular plate with one step variation (Table 7):
)(2,)(2.0,)(1,)(1.0,)(25.0 22110 mrmhmrmhmr ===== .
To demonstrate further the high accuracy of the present exact solution, in Tables 5-7,
ANSYS software package of version 12 and ABAQUS software package of version 6.10
are used to model stepped FGM1 circular/annular plates. The plates are analyzed with a
shell element of type Shell 281 in ANSYS and a solid element in ABAQUS, created on
the basis of the FSDT and the 3D elasticity, respectively. A mesh sensitivity analysis was
carried out to ensure independency of finite element (FE) results from the number of
elements. Results of Tables 5-7 reveal that very good agreement is achieved for the
circular and annular FG plates. But the present results are closer to the FE results on the
basis of the FSDT (Shell 281). This is due to the fact that both methods are based on the
hypothesis of FSDT. It is also noticeable that the difference between the present results
and those obtained by the 3D FE analysis is very small and does not exceed 1% for the
worst case. This closeness is apparent even for higher vibrating modes.
3.2. Results and discussion
-
The present exact procedure may be applied to investigate the effects of various
geometrical and material properties such as step thickness ratios, step locations, the
power law index and different boundary conditions.
3.2.1 Effect of step location on the frequency parameter
Figs. (2a-c) show the variations of the first three frequency parameters versus the step
location 21 / rr= for free, simply supported and clamped circular FGM1 Mindlin plates
(g=1) with one step variation. The step thickness ratio 21 / hh and plate thickness ratio
22 / rh are set to be 3/2 and 0.1, respectively. It is obvious from Figs. (2a-c) that
regardless of the boundary conditions at the plate edges, the step location has severe
influences on the fundamental frequency, especially for high values of . Figs. (2a-c)
prove the fact that the effect of step location on the frequency parameter is more
pronounced for higher vibrating modes and a plate with the higher constraints at its edges
(in the order from free to simply supported to clamped). Fig. 2a shows that as the increase
of the step location , the first three frequency parameters of free stepped circular
FGM1 plate increase slowly to their maximum values and then decrease to the values
corresponding to a free circular FGM1 plate without step variation. The maximum values
of the first, second and third modes are around 0.8, 0.85 and 0.9, respectively. Most of
the time when we use stepped circular plates, it is necessary to alter the natural frequency
by changing the step location . As can be seen, each curve in this figure has one
extrema point which by choosing an appropriate value of , the fundamental frequency
of free stepped circular FG plate attains its maximum value that it can be helpful for
optimal design of FG plates. Figs. (2b-c) reveal that the frequency parameters of clamped
-
and simply supported circular plates increase with the increase the step location . The
reason is that, the stiffness of plate increases with increasing the , leading to the
increase of the frequency parameters.
3.2.2 Effect of the power law index g on the natural frequency
Figs. (3a-b) depict the fundamental natural frequency (Hz) of clamped and simply
supported FG circular plates with one step variation versus the power law index g. The
results are obtained for two different materials as the FGM1 and FGM2. The step
location 21 / rr , the step thickness ratio 21 / hh and the plate thickness ratio 22 / rh are fixed
at 0.5, 5/4 and 0.1, respectively. It is seen that the power law index g has a highly
significant influence on the fundamental natural frequency (Hz) of the plate, especially
for low values of the g (i.e. 0
-
To study the behavior of the frequency parameters against the step thickness ratio ,
all other parameters of plates should be fixed except the . According to Eq. (2), the step
thickness affects on the volume fraction. In order to keep the material parameters of the
plate fixed, it is assumed that the top surface of each step ( 2/1=i
Z ) is the same metal
rich and the bottom one( 2/1=i
Z ) is the same ceramic rich.
The variation of the first three frequency parameters of free, simply supported and
clamped circular FGM1 plates with one step variation versus the step thickness ratio
21 / hh= is shown in Figs. (4a-c). The power law index g, the step location 21 / rr and the
plate thickness ratio 22 / rh are set to be 1, 0.5 and 0.05, respectively. Except for the first
and second modes of the simply supported and clamped FGM1 plates, as the step
thickness ratio enhances, the frequency parameters increase, keeping all other
parameters fixed. The frequency parameters of the first and second modes of the simply
supported and clamped plates increase slowly to their maximum values and then
decrease, indicating that the plate stiffnesses corresponding to the first and second modes
are maximum at the points around 2.2= and 8.1= , respectively , for both the
boundary conditions.
Plots of the first three frequency parameters with respect to the step thickness ratio
322/hh= are shown in Figs. (5a-c) for free, simply supported and clamped circular
FGM1 plates with two step variations when 1=g , 3/1/ 31 =rr , 3/2/ 32 =rr , 1/ 31 =hh
and 15/1/33
=rh . Regardless of boundary constraints and mode numbers, frequency
parameters are considerably increased by increasing the step thickness ratio 2
from
0.5 to 2.5. The effect of the step thickness ratio 2
becomes more pronounced for higher-
-
mode natural frequencies. Such behavior is due to the influence of rotary inertia and
shear deformation.
4. Conclusions
The main objective of this paper was to develop an exact closed form procedure in
solving the free vibration problem of stepped circular and annular FG plates based on the
Mindlin theory. The domain decomposition technique is employed to solve the
eigenfrequency problem of stepped FG plate. Due to the presence of in-plane and out-of-
plane coupling, the five governing complicated partial differential equations of motion
were simultaneously solved by introducing some auxiliary and potential functions. The
accuracy of the present solution is verified by an appropriate FE analysis and checked
with the available literature. It was observed that the proposed procedure yields an exact
closed form solution with an improved accuracy for in-plane and out-of-plane vibration
of stepped FG plates, for the first time. Finally, the influence of different parameters of
the stepped FG plate such as step thickness ratios, step locations and the power law index
on the natural frequencies is studied. Results show that the step parameters (i.e. step
thickness ratios and step locations) play a significant role in the determination of
vibration behavior of the FG plate especially for higher vibrating modes. The merit and
convenience of the present procedure will enable every reader to pursue the various steps
of the solution and, therefore, he/she can easily compute the exact frequency parameters
of stepped circular and annular FG plates.
-
References
[1] A.W. Leissa, Vibrations of Plates, NASA SP-160, USA, 1969.
[2] A.W. Leissa, Recent research in plate vibrations: classical theory, The Shock and
Vibration Digest 9 (1977) 1324.
[3] A.W. Leissa, Plate vibration research, 19761980: classical theory, The Shock and
Vibration Digest 13 (1981) 1122.
[4] A.W. Leissa, Recent studies in plate vibrations: 19811985, Part I: classical theory,
The Shock and Vibration Digest 19 (1987) 1118.
[5] B. Singh, V. Saxena, Transverse vibration of a quarter of a circular plate with variable
thickness. J. Sound Vib. 183(1) (1995) 4967.
[6] R.P. Singh, S.K. Jain, Free asymmetric transverse vibration of polar orthotropic
annular sector plate with thickness varying parabolically in radial direction, Sadhana
Acad Proc. Eng. Sci. 29(5) ( 2004) 41528.
[7] X. Wang, Y. Wang, Free vibration analyses of thin sector plates by the new version of
differential quadrature method. Comput Methods Appl. Mech. Eng. 193(3638) (2004)
395771.
[8] J. Wang, Generalized power series solutions of the vibration of classical circular
plates with variable thickness, J. Sound Vib. 202 (1997) 593599.
[9] D.Y. Chen, B.S. Ren, Finite element analysis of the lateral vibration of thin annular
and circular plates with variable thickness, ASME, Journal of Vibration and Acoustics
120 (1998) 747752.
-
[10] T.Y. Wu, Y.Y. Wang, G.R. Liu, Free vibration analysis of circular plates using
generalized differential quadrature rule, Comput Methods Appl. Mech. Eng. 191 (2002)
536580
[11] T.Y. Wu, G.R. Liu, Free vibration analysis of circular plates with variable thickness
by the generalized differential quadrature rule, Int. J. Solids Struct. 38 (2001) 79677980.
[12] A.M. Al-Jumaily, K. Jameel, Influence of the poisson ratio on the natural
frequencies of stepped-thickness circular plate, J. Sound Vib. 234(5) (2000) 881-94.
[13] B. Liang, S. Zhang, D. Chen, Natural frequencies of circular annular plates with
variable thickness by a new method, Int. J. Pressure Vessels Piping 84 (2007) 29397.
[14] Y. Xiang, L. Zhang, Free vibration analysis of stepped circular Mindlin plates, J.
Sound Vib. 280 (2005) 63355.
[15] W.H. Duan, C.M. Wang, C.Y. Wang, Modification of fundamental vibration modes
of circular plates with free edges, J. Sound Vib. 317 (2008) 70915.
[16] L. Hang, C.M. Wang, T.Y.Wu, Exact vibration results for stepped circular plates
with free edges, Int. J. Mech. Sci. 47 (2005) 122448.
[17] Sh. Hosseini-Hashemi, M. Eshaghi, H. Rokni Damavandi Taher, M. Fadaie, Exact
closed-form frequency equations for thick circular plates using a third-order shear
deformation theory, J. Sound Vib. 329 (2010) 338296.
[18] J.H. Kang, A.W. Leissa, Three-dimensional vibrations of thick, linearly tapered,
annular plates, J. Sound Vib. 217 (1998) 92744.
[19] J.H. Kang, Three-dimensional vibration analysis of thick, circular and annular plates
with nonlinear thickness variation, Computers and Structures 81 (2003) 166375.
-
[20] Sh. Hosseini-Hashemi, H. Rokni Damavandi Taher, H. Akhavan, Vibration analysis
of radially FGM sectorial plates of variable thickness on elastic foundations, Composite
Structures 92 (2010) 173443.
[21] Sh. Hosseini-Hashemi, M. Fadaee, M. Eshaghi, A novel approach for in-plane/out-
of-plane frequency analysis of functionally graded circular/annular plates, Int. J. Mech.
Sci. 52 (2010) 102535.
[22] U.S. Gupta, R. Lal, S. Sharma, Vibration of non-homogeneous circular Mindlin
plates with variable thickness, J. Sound Vib. 302 (2007) 117.
[23] E. Efraim, M. Eisenberger, Exact vibration analysis of variable thickness thick
annular isotropic and FGM plates, J. Sound Vib. 299 (2007) 72038.
[24] V. Tajeddini, A. Ohadi, A, Sadighi, Three-dimensional free vibration of variable
thickness thick circular and annular isotropic and functionally graded plates on Pasternak
foundation, Int. J. Mech. Sci. 53 (2011) 30008.
[25] F. Tornabene, Free vibration analysis of functionally graded conical, cylindrical
shell and annular plate structures with a four-parameter power- law distribution. Comput.
Methods Appl. Mech. Eng. 198 (2009) 291135.
[26] F. Tornabene, E. Viola, D. J. Inman, 2-D differential quadrature solution for
vibration analysis of functionally graded conical, cylindrical shell and annular plate
structures. J. Sound Vib. 328 (2009) 259290.
[27] J. Alan, D. Hui-Hui, Handbook of Mathematical Formulas and Integrals, Fourth Ed.,
Academic Press, New York, 2008.
-
Figures Caption
Fig. 1. Geometry of a stepped annular FG plate.
Fig. 2. Variation of the first three frequency parameters b versus the step location
21 / rr= for (a) free (b) simply supported (c) clamped circular FGM1 Mindlin plates
with one step variation.
Fig. 3. Variation of the fundamental frequency parameter b versus the power law
index g for clamped and simply supported (a) FGM1, (b) FGM2 circular plates with
one step variation.
Fig. 4. Variation of the first three frequency parameters b versus the step thickness
ratio 21 / hh=t for (a) free (b) simply supported (c) clamped circular FGM1 Mindlin
plates with one step variation.
Fig. 5. Variation of the first three frequency parameters b versus the step thickness
ratio 322
/hh=t for (a) free (b) simply supported (c) clamped circular FGM1 Mindlin
plates with two step variations.
-
Fig. 1. Geometry of a stepped annular FG plate
-
(a) (b)
`(c)
Fig. 2. Variation of the first three frequency parameters b versus the step location 21 / rr= for (a) free (b) simply
supported (c) clamped circular FGM1 Mindlin plates with one step variation.
b
0.2 0.4 0.6 0.8 1
4
6
8
10
12
14
First mode
Second mode
Third mode
b
0.2 0.4 0.6 0.8 1
5
10
15
20
25
30 First mode
Second mode
Third mode
b
0.2 0.4 0.6 0.8 15
10
15
20
25
30
35 First mode
Second mode
Third mode
-
(a) (b)
Fig. 3. Variation of the fundamental frequency parameter b versus the power law index g for clamped and simply
supported (a) FGM1, (b) FGM2 circular plates with one step variation.
g
w
0 20 40 60
100
150
200
250
Simply Support
Clamped
g
w0 20 40 60
60
80
100
120
140
Simply Support
Clamped
-
(a) (b)
`(c)
Fig. 4. Variation of the first three frequency parameters b versus the step thickness ratio 21 / hh=t for (a) free (b)
simply supported (c) clamped circular FGM1 Mindlin plates with one step variation.
t
b
0.5 1 1.5 2 2.5
4
6
8
10
12
Third mode
Second mode
First mode
t
b0.5 1 1.5 2 2.5
0
5
10
15
20
25
Third mode
Second mode
First mode
t
b
0.5 1 1.5 2 2.55
10
15
20
25
30
35Third mode
Second mode
First mode
-
(a) (b)
`(c)
Fig. 5. Variation of the first three frequency parameters b versus the step thickness ratio 322
/hh=t for (a) free (b)
simply supported (c) clamped circular FGM1 Mindlin plates with two step variations.
t2
b
0.5 1 1.5 2 2.52
4
6
8
10
12
14
16
18
Third mode
Second mode
First mode
t2b
0.5 1 1.5 2 2.5
5
10
15
20
25
Third mode
Second mode
First mode
t2
b
0.5 1 1.5 2 2.5
5
10
15
20
25
30
Third mode
Second mode
First mode
-
Tab
les
Cap
tion
Tab
le 1
. M
ater
ial
pro
per
ties
of
the
use
d F
G p
late
.
Tab
le 2
. C
om
par
ison s
tud
y o
f fi
rst
ten n
atura
l fr
equen
cies
(H
z) f
or
a C
-C F
GM
1 a
nnula
r p
late
()
(1
,)
(2.
01
0m
rm
r=
=).
Tab
le 3
. C
om
par
ison s
tud
y o
f fi
rst
ten f
requen
cy p
aram
eter
s b
for
isotr
op
ic c
ircu
lar
pla
te w
ith o
ne
step
var
iati
on (
1.0
/,
5.0
/2
22
1=
=r
hr
r).
Tab
le 4
. C
om
par
ison s
tud
y o
f fi
rst
ten f
requen
cy p
aram
eter
s b
for
isotr
op
ic c
ircu
lar
pla
te w
ith t
wo s
tep
var
iati
ons
(
1.0
/,
3/2
/,
3/1
/3
33
23
1=
==
rh
rr
rr
).
Tab
le 5
. C
om
par
ison s
tud
y o
f fi
rst
ten n
atura
l fr
equen
cies
(H
z) f
or
FG
M1 c
ircu
lar
pla
te w
ith o
ne
step
var
iati
on
(
1,)
(2
,)
(1.
0,
)(
1,
)(
2.0
22
11
==
==
=g
mr
mh
mr
mh
).
Tab
le 6
. C
om
par
ison s
tud
y o
f fi
rst
ten n
atura
l fr
equen
cies
(H
z) f
or
FG
M1 c
ircu
lar
pla
te w
ith t
wo s
tep
var
iati
ons
(
1,
)(
4,
)(
16
.0
,)
(2
,)
(2.
0,
)(
1,
)(
24
.0
33
22
11
==
==
==
=g
mr
mh
mr
mh
mr
mh
).
Tab
le 7
. C
om
par
ison s
tud
y o
f fi
rst
ten n
atura
l fr
equen
cies
(H
z) f
or
FG
M1 a
nnula
r p
late
wit
h o
ne
step
var
iati
on (
1,)
(2
,)
(2.
0,
)(
1,
)(
1.0
,)
(25
.0
22
11
0=
==
==
=g
mr
mh
mr
mh
mr
).
-
Tab
le 1
. M
ater
ial
pro
per
ties
of
the
use
d F
G p
late
.
FG
Mat
eria
l P
rop
erti
es
E
m (
GP
a)
Ec
(GP
a)
mr
(Kg/m
3)
cr
(Kg/m
3)
n
FG
M1
70
380
2700
38
00
0.3
FG
M2
70
205
2700
60
50
0.3
-
Tab
le 2
. C
om
par
ison s
tud
y o
f fi
rst
ten n
atura
l fr
equen
cies
(H
z) f
or
a C
-C F
GM
1 a
nnula
r p
late
()
(1
,)
(2.
01
0m
rm
r=
=).
M
od
e nu
mb
er
(0,1
) (1
,1)
(2,1
) (3
,1)
(4,1
) (5
,1)
(0,2
) (1
,2)
(2,2
) (6
,1)
h=
0.0
1
Pre
sen
t 1
27.1
10
132.5
82
153.5
51
196.0
14
257.9
18
33
3.4
22
35
1.1
51
36
0.4
27
39
0.5
71
41
9.0
36
E
xac
t [2
1]
12
7.1
10
132.5
82
153.5
51
196.0
14
257.9
18
33
3.4
22
35
1.1
51
36
0.4
27
39
0.5
71
41
9.0
36
F
EM
(3D
) 1
26.0
90
131.5
00
152.3
20
194.4
60
255.9
40
33
1.1
10
34
8.3
70
35
7.5
40
38
7.4
90
41
6.5
70
M
od
e nu
mb
er
(0,1
) (1
,1)
(2,1
) (3
,1)
(4,1
) (5
,1)
(0,2
) (1
,2)
(2,2
) (0
,3)
h=
0.1
P
rese
nt
11
52
.56
1195.9
8
1379.0
3
1755.1
5
2274.9
4
28
70
.38
28
91
.44
29
61
.24
31
96
.58
34
75
.86
Ex
act
[21]
11
52
.56
1195.9
8
1379.0
3
1755.1
5
2274.9
4
28
70
.38
28
91
.44
29
61
.24
31
96
.58
34
75
.86
FE
M(3
D)
11
52
.10
1195.4
0
1377.3
0
1752.0
0
2272.5
0
28
71
.50
29
04
.60
29
74
.90
32
10
.00
34
75
.30
M
od
e nu
mb
er
(0,1
) (1
,1)
(2,1
) (3
,1)
(0,2
) (4
,1)
(1,2
) (0
,3)
(1,3
) (5
,1)
h=
0.2
P
rese
nt
18
66
.04
1936.2
0
2247.4
9
2840.2
6
3467.3
7
35
86
.08
38
47
.22
41
47
.27
42
70
.61
43
88
.59
Ex
act
[21]
18
66
.04
1936.2
0
2247.4
9
2840.2
6
3467.3
7
35
86
.08
38
47
.22
41
47
.27
42
70
.61
43
88
.59
FE
M(3
D)
18
85
.60
1955.4
0
2266.2
0
2864.2
0
3465.0
0
36
23
.70
38
53
.60
42
24
.50
43
46
.30
44
46
.10
-
Tab
le 3
. C
om
par
ison s
tud
y o
f fi
rst
ten f
requen
cy p
aram
eter
s b
for
isotr
op
ic c
ircu
lar
pla
te w
ith o
ne
step
var
iati
on (
1.0
/,
5.0
/2
22
1=
=r
hr
r).
B. C
s
21/h
h
Met
ho
d
The
freq
uen
cy p
aram
eter
s b
wit
h t
he
corr
esp
ondin
g m
ode
num
ber
s (m
,n)
Fre
e 0.5
Pre
sen
t
4.3
2805
(2,1
)
6.7
3006
(0,1
)
11.2
017
(3,1
)
14.6
336
(4,1
)
20.1
792
(1,1
)
24
.36
12
(5,1
)
27
.07
89
(2,2
)
30
.88
34
(0,2
)
40
.90
33
(6,1
)
41
.85
69
(3,2
)
E
xac
t [1
4]
4.3
2805
(2,1
)
6.7
3006
(0,1
)
11.2
017
(3,1
)
14.6
336
(4,1
)
20.1
792
(1,1
)
24
.36
12
(5,1
)
27
.07
89
(2,2
)
30
.88
34
(0,2
)
40
.90
33
(6,1
)
41
.85
69
(3,2
)
1.5
Pre
sen
t
6.8
8138
(2,1
)
11.4
342
(0,1
)
13.3
881
(3,1
)
21.6
542
(4,1
)
22.2
602
(1,1
)
31
.77
25
(5,1
)
36
.10
61
(2,2
)
41
.06
24
(0,2
)
43
.54
79
(6,1
)
52
.79
97
(3,2
)
E
xac
t [1
4]
6.8
8138
(2,1
)
11.4
342
(0,1
)
13.3
881
(3,1
)
21.6
542
(4,1
)
22.2
602
(1,1
)
31
.77
25
(5,1
)
36
.10
61
(2,2
)
41
.06
24
(0,2
)
43
.54
79
(6,1
)
52
.79
97
(3,2
)
2
Pre
sen
t
8.5
6567
(2,1
)
13.0
908
(0,1
)
14.5
691
(3,1
)
22.3
198
(4,1
)
22.7
786
(1,1
)
32
.12
22
(5,1
)
38
.43
14
(2,2
)
43
.72
44
(6,1
)
46
.02
70
(0,2
)
56
.82
97
(7,1
)
E
xac
t [1
4]
8.5
6567
(2,1
)
13.0
908
(0,1
)
14.5
691
(3,1
)
22.3
198
(4,1
)
22.7
786
(1,1
)
32
.12
22
(5,1
)
38
.43
14
(2,2
)
43
.72
44
(6,1
)
46
.02
70
(0,2
)
56
.82
97
(7,1
)
Har
d S
imp
ly-
Sup
port
ed
0.5
Pre
sen
t
4.0
1011
(0,1
)
10.7
583
(1,1
)
20.8
502
(0,2
)
21.0
157
(2,1
)
33.4
553
(3,1
)
35
.34
08
(1,2
)
47
.99
46
(4,1
)
49
.73
70
(0,3
)
50
.39
17
(2,2
)
63
.86
92
(1,3
)
E
xac
t [1
4]
4.0
1011
(0,1
)
10.7
583
(1,1
)
20.8
502
(0,2
)
21.0
157
(2,1
)
33.4
553
(3,1
)
35
.34
08
(1,2
)
47
.99
46
(4,1
)
49
.73
70
(0,3
)
50
.39
17
(2,2
)
63
.86
92
(1,3
)
1.5
Pre
sen
t
5.6
8854
(0,1
)
14.4
923
(1,1
)
26.9
433
(2,1
)
32.6
687
(0,2
)
41.1
944
(3,1
)
52
.28
97
(1,2
)
55
.80
52
(4,1
)
70
.19
48
(2,2
)
70
.93
04
(5,1
)
75
.19
29
(0,3
)
E
xac
t [1
4]
5.6
8854
(0,1
)
14.4
923
(1,1
)
26.9
433
(2,1
)
32.6
687
(0,2
)
41.1
944
(3,1
)
52
.28
97
(1,2
)
55
.80
52
(4,1
)
70
.19
48
(5,1
)
70
.93
04
(2,2
)
75
.19
29
(0,3
)
2
Pre
sen
t
5.9
4088
(0,1
)
14.3
961
(1,1
)
29.5
769
(2,1
)
37.4
877
(0,2
)
45.1
643
(3,1
)
56
.88
33
(1,2
)
59
.27
01
(4,1
)
73
.04
26
(2,2
)
73
.40
61
(5,1
)
78
.98
00
(0,3
)
E
xac
t [1
4]
5.9
4088
(0,1
)
14.3
961
(1,1
)
29.5
769
(2,1
)
37.4
877
(0,2
)
45.1
643
(3,1
)
56
.88
33
(1,2
)
59
.27
01
(4,1
)
73
.04
26
(5,1
)
73
.40
61
(2,2
)
78
.98
00
(0,3
)
Cla
mp
ed
0.5
Pre
sen
t
9.9
7265
(0,1
)
16.9
103
(1,1
)
25.6
426
(0,2
)
27.2
909
(2,1
)
40.1
942
(3,1
)
40
.96
60
(1,2
)
55
.17
54
(4,1
)
57
.32
16
(2,2
)
59
.20
26
(0,3
)
71
.89
33
(5,1
)
Ex
act
[14]
9.9
7265
(0,1
)
16.9
103
(1,1
)
25.6
426
(0,2
)
27.2
909
(2,1
)
40.1
942
(3,1
)
40
.96
60
(1,2
)
55
.17
54
(4,1
)
57
.32
16
(2,2
)
59
.20
26
(0,3
)
71
.89
33
(5,1
)
1.5
Pre
sen
t
10
.6655
(0,1
)
21.7
095
(1,1
)
35.0
502
(2,1
)
41.5
770
(0,2
)
50.3
801
(3,1
)
62
.72
43
(1,2
)
65
.79
79
(4,1
)
81
.13
64
(5,1
)
81
.53
27
(2,2
)
86
.57
88
(0,3
)
E
xac
t [1
4]
10
.6655
(0,1
)
21.7
095
(1,1
)
35.0
502
(2,1
)
41.5
770
(0,2
)
50.3
801
(3,1
)
62
.72
43
(1,2
)
65
.79
79
(4,1
)
81
.13
64
(5,1
)
81
.53
27
(2,2
)
86
.57
88
(0,3
)
2
Pre
sen
t
11
.0254
(0,1
)
21.6
130
(1,1
)
37.5
335
(2,1
)
46.7
177
(0,2
)
55.0
429
(3,1
)
69
.19
43
(1,2
)
70
.34
28
(4,1
)
84
.53
75
(5,1
)
85
.53
28
(2,2
)
90
.79
40
(0,3
)
E
xac
t [1
4]
11
.0254
(0,1
)
21.6
130
(1,1
)
37.5
335
(2,1
)
46.7
177
(0,2
)
55.0
429
(3,1
)
69
.19
43
(1,2
)
70
.34
28
(4,1
)
84
.53
75
(5,1
)
85
.53
28
(2,2
)
90
.79
40
(0,3
)
-
Tab
le 4
. C
om
par
ison s
tud
y o
f fi
rst
ten f
requen
cy p
aram
eter
s b
for
isotr
op
ic c
ircu
lar
pla
te w
ith t
wo s
tep v
aria
tions
(1.
0/
,3/
2/
,3/
1/
33
32
31
==
=r
hr
rr
r).
B. C
s
32
/hh
3
1/h
h
Met
ho
d
The
freq
uen
cy p
aram
eter
s b
wit
h t
he
corr
esp
ondin
g m
ode
num
ber
s (m
,n)
Fre
e
1.5
2
Pre
sen
t
8.6
4131
(2,1
)
14.2
970
(0,1
)
15.7
309
(3,1
)
24.2
700
(4,1
)
26.7
991
(1,1
)
34
.29
76
(5,1
)
41
.71
96
(2,2
)
45
.75
80
(6,1
)
47
.33
91
(0,2
)
57
.37
19
(3,2
)
Exac
t [1
4]
8.6
4131
(2,1
)
14.2
970
(0,1
)
15.7
309
(3,1
)
24.2
700
(4,1
)
26.7
991
(1,1
)
34
.29
76
(5,1
)
41
.71
96
(2,2
)
45
.75
80
(6,1
)
47
.33
91
(0,2
)
57
.37
19
(3,2
)
2
3
Pre
sen
t
12.4
084
(2,1
)
18.9
759
(0,1
)
19.5
384
(3,1
)
27.5
316
(4,1
)
30.7
345
(1,1
)
36
.85
59
(5,1
)
45
.92
72
(2,2
)
47
.67
12
(6,1
)
52
.98
87
(0,2
)
59
.95
20
(7,1
)
Exac
t [1
4]
12.4
084
(2,1
)
18.9
759
(0,1
)
19.5
384
(3,1
)
27.5
316
(4,1
)
30.7
345
(1,1
)
36
.85
59
(5,1
)
45
.92
72
(2,2
)
47
.67
12
(6,1
)
52
.98
87
(0,2
)
59
.95
20
(7,1
)
2/3
1/3
Pre
sen
t
3.9
4684
(2,1
)
5.9
6010
(0,1
)
10.4
301
(3,1
)
13.0
014
(1,1
)
19.0
926
(4,1
)
23
.28
01
(0,2
)
24
.35
79
(2,2
)
29
.64
03
(5,1
)
35
.38
55
(1,2
)
38
.14
28
(3,2
)
Exac
t [1
4]
3.9
4684
(2,1
)
5.9
6010
(0,1
)
10.4
301
(3,1
)
13.0
014
(1,1
)
19.0
926
(4,1
)
23
.28
01
(0,2
)
24
.35
79
(2,2
)
29
.64
03
(5,1
)
35
.38
55
(1,2
)
38
.14
28
(3,2
)
Har
d S
imp
ly-
Supp
ort
ed
1.5
2
Pre
sen
t
6.6
1467
(0,1
)
16.3
417
(1,1
)
29.7
116
(2,1
)
36.4
009
(0,2
)
43.9
133
(3,1
)
56
.31
20
(1,2
)
59
.55
87
(4,1
)
76
.36
19
(5,1
)
78
.42
03
(2,2
)
84
.52
69
(0,3
)
E
xac
t [1
4]
6.6
1467
(0,1
)
16.3
417
(1,1
)
29.7
116
(2,1
)
36.4
009
(0,2
)
43.9
133
(3,1
)
56
.31
20
(1,2
)
59
.55
87
(4,1
)
76
.36
19
(5,1
)
78
.42
03
(2,2
)
84
.52
69
(0,3
)
2
3
Pre
sen
t
7.4
4372
(0,1
)
16.9
413
(1,1
)
33.1
898
(2,1
)
42.3
607
(0,2
)
49.1
513
(3,1
)
65
.24
31
(1,2
)
66
.37
87
(4,1
)
84
.10
66
(5,1
)
90
.03
04
(2,2
)
97
.05
42
(0,3
)
Exac
t [1
4]
7.4
4372
(0,1
)
16.9
413
(1,1
)
33.1
898
(2,1
)
42.3
607
(0,2
)
49.1
513
(3,1
)
65
.24
31
(1,2
)
66
.37
87
(4,1
)
84
.10
66
(5,1
)
90
.03
04
(2,2
)
97
.05
42
(0,3
)
2/3
1/3
Pre
sen
t
3.6
7234
(0,1
)
9.8
4061
(1,1
)
19.2
783
(2,1
)
19.6
735
(0,2
)
30.0
635
(1,2
)
31
.01
72
(3,1
)
40
.16
03
(0,3
)
44
.33
26
(2,2
)
44
.80
90
(4,1
)
59
.70
72
(1,3
)
Exac
t [1
4]
3.6
7234
(0,1
)
9.8
4061
(1,1
)
19.2
783
(2,1
)
19.6
735
(0,2
)
30.0
635
(1,2
)
31
.01
72
(3,1
)
40
.16
03
(0,3
)
44
.33
26
(2,2
)
44
.80
90
(4,1
)
59
.70
72
(1,3
)
Cla
mp
ed
1.5
2
Pre
sen
t
11.3
152
(0,1
)
23.7
479
(1,1
)
38.6
283
(2,1
)
46.1
750
(0,2
)
53.5
294
(3,1
)
66
.55
35
(1,2
)
69
.50
23
(4,1
)
86
.62
46
(5,1
)
88
.65
46
(2,2
)
94
.71
15
(0,3
)
E
xac
t [1
4]
11.3
152
(0,1
)
23.7
479
(1,1
)
38.6
283
(2,1
)
46.1
750
(0,2
)
53.5
294
(3,1
)
66
.55
35
(1,2
)
69
.50
23
(4,1
)
86
.62
46
(5,1
)
88
.65
46
(2,2
)
94
.71
15
(0,3
)
2
3
Pre
sen
t
12.4
167
(0,1
)
24.9
392
(1,1
)
42.0
639
(2,1
)
51.7
205
(0,2
)
58.5
030
(3,1
)
74
.90
68
(1,2
)
76
.31
20
(4,1
)
94
.92
19
(5,1
)
10
0.8
97
(2,2
)
10
8.4
13
(0,3
)
E
xac
t [1
4]
12.4
167
(0,1
)
24.9
392
(1,1
)
42.0
639
(2,1
)
51.7
205
(0,2
)
58.5
030
(3,1
)
74
.90
68
(1,2
)
76
.31
20
(4,1
)
94
.92
19
(5,1
)
10
0.8
97
(2,2
)
10
8.4
13
(0,3
)
2/3
1/3
Pre
sen
t
9.6
4461
(0,1
)
15.9
370
(1,1
)
25.5
801
(0,2
)
25.6
633
(2,1
)
36.9
919
(1,2
)
37
.61
63
(3,1
)
46
.12
07
(0,3
)
50
.81
87
(2,2
)
51
.64
33
(4,1
)
65
.03
09
(1,3
)
E
xac
t [1
4]
9.6
4461
(0,1
)
15.9
370
(1,1
)
25.5
801
(0,2
)
25.6
633
(2,1
)
36.9
919
(1,2
)
37
.61
63
(3,1
)
46
.12
07
(0,3
)
50
.81
87
(2,2
)
51
.64
33
(4,1
)
65
.03
09
(1,3
)
-
Tab
le 5
. C
om
par
ison s
tud
y o
f fi
rst
ten n
atura
l fr
equen
cies
(H
z) f
or
FG
M1 c
ircu
lar
pla
te w
ith o
ne
step
var
iati
on
(1
,)(
2,
)(
1.0
,)
(1
,)
(2.
02
21
1=
==
==
gm
rm
hm
rm
h).
B.
Cs
M
od
e nu
mb
er (
m,n
) (2
,1)
(0,1
) (3
,1)
(4,1
) (1
,1)
(5,1
) (2
,2)
(6,1
) (0
,2)
(7,1
)
Fre
e P
rese
nt
83.5
43
128.2
89
146.3
98
227.5
83
230.4
16
33
1.6
64
39
5.6
39
45
7.7
97
47
7.2
22
60
4.3
37
F
EM
(F
SD
T)
83.5
78
128.3
40
146.4
60
227.6
70
230.5
10
33
1.8
00
39
5.7
90
45
8.0
00
47
7.4
00
60
4.6
60
F
EM
(3
D)
82.7
73
125.6
58
145.1
56
225.1
39
226.5
25
33
1.0
18
39
2.8
33
45
7.6
37
47
7.7
37
60
4.6
96
M
od
e nu
mb
er (
m,n
) (0
,1)
(1,1
) (2
,1)
(0,2
) (3
,1)
(1,2
) (4
,1)
(5,1
) (2
,2)
(0,3
)
Soft
-Sim
ply
P
rese
nt
58.0
84
144.0
63
297.9
91
382.9
25
468.4
89
60
6.8
05
62
8.4
76
79
0.8
95
79
6.9
88
86
7.9
57
Supp
ort
ed
FE
M (
FS
DT
) 58.1
08
144.1
20
298.1
10
383.0
70
468.6
50
60
7.0
00
62
8.6
90
79
1.1
70
79
7.2
30
86
8.2
20
F
EM
(3
D)
57.3
17
142.3
86
297.9
23
383.9
24
469.6
95
60
3.8
36
63
0.5
83
79
0.8
94
79
4.8
23
86
0.8
99
M
od
e nu
mb
er (
m,n
) (0
,1)
(1,1
) (2
,1)
(0,2
) (3
,1)
(1,2
) (4
,1)
(5,1
) (2
,2)
(0,3
)
Cla
mp
ed
Pre
sent
110.6
29
223.2
79
392.7
35
493.5
37
594.2
90
76
7.1
16
78
2.0
61
95
8.9
38
97
8.9
60
10
44
.08
3
F
EM
(F
SD
T)
110.6
70
223.3
60
392.8
70
493.7
10
594.4
80
76
7.3
30
78
2.2
90
95
9.2
20
97
9.2
10
10
44
.30
0
F
EM
(3
D)
109.8
80
220.4
65
392.1
32
495.4
84
596.7
16
76
7.3
74
78
5.8
48
96
3.8
67
97
2.7
48
10
33
.770
-
Tab
le 6
. C
om
par
ison s
tud
y o
f fi
rst
ten n
atura
l fr
equen
cies
(H
z) f
or
FG
M1
cir
cula
r pla
te w
ith t
wo s
tep
var
iati
ons
(
1,
)(
4,
)(
16
.0
,)
(2
,)
(2.
0,
)(
1,
)(
24
.0
33
22
11
==
==
==
=g
mr
mh
mr
mh
mr
mh
).
B.
Cs
M
od
e nu
mb
er (
m,n
) (2
,1)
(0,1
) (3
,1)
(1,1
) (4
,1)
(5,1
) (2
,2)
(0,2
) (6
,1)
(3,2
)
Fre
e P
rese
nt
24.2
81
40.8
55
52.3
16
85.1
90
85.5
74
12
8.6
74
14
5.0
38
16
2.6
11
18
0.2
41
21
3.5
92
F
EM
(F
SD
T)
24.2
72
40.8
67
50.4
17
85.2
13
85.2
59
12
8.6
60
14
4.6
30
16
2.6
60
18
0.2
90
21
3.2
60
F
EM
(3
D)
24.2
95
40.7
06
50.4
03
85.0
51
85.2
51
12
8.6
78
14
4.5
24
16
2.5
38
18
0.3
20
21
3.3
84
M
od
e nu
mb
er (
m,n
) (0
,1)
(1,1
) (2
,1)
(0,2
) (3
,1)
(1,2
) (4
,1)
(2,2
) (5
,1)
(0,3
)
Soft
-Sim
ply
Supp
ort
ed
Pre
sent
21.0
09
56.0
09
106.1
62
126.2
12
161.7
21
20
4.8
98
22
4.4
99
29
1.6
48
29
4.7
85
31
5.8
28
FE
M (
FS
DT
) 2
1.0
15
56.0
24
105.1
70
126.2
50
161.0
80
20
4.9
50
22
4.3
20
29
1.6
70
29
4.8
10
31
5.9
00
F
EM
(3
D)
20.9
34
55.9
91
105.1
06
126.0
39
161.0
17
20
5.2
65
22
4.2
75
29
1.9
88
29
4.6
26
31
5.4
24
M
od
e nu
mb
er (
m,n
) (0
,1)
(1,1
) (2
,1)
(0,2