free vibration response of functionally graded carbon...
TRANSCRIPT
1
Free vibration response of functionally graded carbon nanotubes
double curved shell and panel with piezoelectric layers in a thermal
environment
Siros Khorshidi a*
, Saeed Saber-Samandari b, Manouchehr Salehi
a
a
Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran.
b New Technologies Research Center, Amirkabir University of Technology, Tehran, Iran
Corresponding author:
Mr. Siros khorshidi
Tel: 0098 9367522833
Email:[email protected]
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Abstract
This paper presents free vibration of the double-curved shells and panels with piezoelectric
layers in a thermal environment. Vibration characteristics of elliptical, spherical, cycloidal, and
toro circular shells of revolution are studied in detail. These structures are made of a based
carbon nanotubes (CNTs) core and piezoelectric layers at the upper and lower surfaces. It is
supposed that temperature changes linearly through the thickness direction. Reissner- Mindlin
and the first order shear deformation (FSDT) theories are implemented to derive the governing
equations of the considered structures. The distribution of nanotubes is assumed to be linear
along the thickness direction. For solving the equation, the General Differential Quadrature
(GDQ) method is implemented to investigate the dynamic behavior of the structures. Finally, the
effects of the boundary conditions, the thickness of piezoelectric layers, the functional
distribution of CNTs, thermal environment and kinds of the circuit (opened-circuit and closed-
circuit) are analyzed. Eigenvalue system is solved to obtain natural frequencies. It is delineated
that the obtained fundamental frequency by the closed -circuit is smaller than those obtained by
the opened-circuit. Another interesting result is that the natural frequency is decreased by
increasing temperature.
Keywords: Vibration; Piezoelectric layer; CNT; Thermal analysis; Mindlin–Reissner theory
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Nomenclature
value of meridional direction p piezoelectric
value of circumferential direction c core
z value of thickness direction ij thermal expansion
cz value of piezoelectric thickness direction T temperature Gradient
h thickness of the structures *V total volume fraction
ph thickness of piezoelectric i efficiency parameter
bR distance between geometric axis and revolution axis poisson coefficient
a semi major axis density
b semi minor axis E ij young modolus
R radious in meridional direction totU strain energy
R radious in circumferential direction totK kinetic energy
0R harizontal radious totV potential energy
r c radius of the produce circle of cycloid curve Ii moment of inertia
U displacement in meridional direction c ij, cpij
weighting coeficients
V displacement in circumferential direction
W displacement in thickness direction
u displacement of reference plane in meridional direction
v displacement of reference plane in circumferential direction
w displacement of reference plane in thickness direction
rotation around axis M Legendre polynomial
rotation around axis M mass matrix
iA Lame´ parameters K stiffness matrix
N force resultant curvature
ijA extensional stiffness M moment resultant
ijB bending-extensional coupling stiffnesses ijD bending stiffness
D i electrical displacements T thermal
ijp pyroelectric property ijk dielectric permittivity
ije piezoelectric stress constant is the electric potential
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1.Introduction
Shells are three-dimensional structures constrained by two curved surfaces, with unique
mechanical properties. Generally, these structures are divided into three groups such as
thick, moderately thick, and thin shells [1]. Because of their excellent and unique
mechanical characteristics, they are implemented in different applications including
submarines, storages, tanks, automobiles, airplanes, etc [2]. One of the investigation
aspects of shells is analyzing their vibration characteristics. Therefore, considering their
extensive applications in different mechanical and civil structures, it is vital to develop an
accurate and reliable dynamical model for shells. Not only shells can be assumed as the
3D body, but also they can behave as 2D structures. In former condition, the governing
equations are derived using the 3D elasticity theory, and for the latter, the shell theory is
implemented. In the shell theory (2D-model), the mid-surface of a structure is considered
as a reference plane, and the system of equations are derived based on the assumed
reference plane. It is noticed that in the present study, we used this theory to extract the
governing equations. In this study, piezoelectric layers are embedded in the upper and
lower surfaces. As a result, it is necessary to provide some applications of piezoelectric in
industries. One of their practical uses is that they can be used as power generation in
automobiles. When an automobile’s tires rotate in roads, the amount of energy goes
waste. This kind of energy can be implemented as the required power for automobile
electronic devices. It is noticed that piezoelectric materials can be used to generate
electrical energy when it is subjected to mechanical strain [3]. Another application which
we can refer to is the tire pressure sensor. It is placed inside a wheel, and by measuring
the amount of force which is imported to the piezoelectric disc, it can be an accurate
indicator for regulating the tire pressure. As we observe the characteristic of piezoelectric
material, which converts the electrical energy to the mechanical kind and vice versa,
provides a potential to reap its benefit [3]. In recent years, there has been a great effort
toward modeling and analyzing the static and dynamic properties of the moderately thick
shells. For example, Tornabene et al. [4] probed the free vibration of anisotropic double
shells using two-dimensional Differential Quadrature (DQ) method. They implemented
the first order shear deformation theory and examined the vibration behavior of the
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moderately thick structures. Viola et al. [5] investigated the static analysis of double
curved shells and panels with higher order shear deformation theory. To achieve this
purpose, they discretized the governing equations of the structure using the Generalized
Differential Quadrature (GDQ) method. Shooshtari et al. [6] used single-mode Galerkin
method to study large amplitude vibration of magneto-electro-elastic curved panels. In
this research, they implemented Donnell shell theory and Gauss's lows for electrostatics
and magnetostatics to derive the governing equations. And, finally, they examined the
first mode of vibration of this structure. Pang et al. [7] implemented a semi-analytical
method to analyze the free vibration of the spherical-cylindrical-spherical shell. They
utilized the Flugge thin shell theory to extract governing equations. For approximating
displacement function, unified Jacobi polynomials and Fourier series were used. Finally,
the results were compared with those which were obtained from the finite element
method (FEM). Rout et al. [8] investigated the free vibration analysis of graphene-
reinforced shell and panels in a thermal environment. In this research, the governing
equations of four type of double curved shells were solved with FEM. Also, Materials
were defined based on the Halpin–Tsai approach to show their dependence on
temperature. Finally, the results were contrasted by other studies to confirm their credits.
Pang et al. [9] analyzed free vibration of the double curved shell of revolution with a
semi-analytical method. The displacements along the revolution axis were approximated
with the Jacobi polynomials, and the Fourier series were used to indicate the movement
along the circumferential direction. The Rayleigh–Ritz method was implemented to
obtain natural frequencies. Awrejcewicz et al. [10] studied linear and nonlinear free
vibration of laminated functionally graded shallow shells. They assumed that these
structures were made of FGM based on power law and varied along the thickness
direction. The first shear deformation theory was implemented to define the displacement
field. For solving equations, the Ritz and the R-functions method were utilized together.
Fang et al. [11] conducted a study that Goldenveizer-Novozhilov shell theory, thin plate
theory, and electro-elastic surface theory were applied to investigate nano-sized
piezoelectric double-shell structures. In this research, the surface energy effect was
considered, and the government equation was solved using the Rayleigh-Ritz method.
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Zhou et al. [12] implemented an accurate approach to analyze the free vibration of
piezoelectric fiber-reinforced composite cylindrical shells in a thermal environment. They
assumed that the composite materials were made in two ways; first, each ply had a
regular distribution, and second, the reinforcements were graded functionally through the
thickness direction. The governing equations were extracted by the Hamilton method,
and for solving them, the researchers used an analytical approach. Mallek et al. [13]
analyzed the electromechanical behavior of composite shell structures with embedded
piezoelectric layers. For solving the equations, they used the 3D- shell model based on
discrete double directors shell elements. It is noticed that they got utilized the third shear
deformation theory as the displacement field. Akbari Alashti et al. [14] studied the
thermo-elastic analysis of a functionally graded double curved shell. They assumed that
the material varied through the thickness direction based on power law, and also the
piezoelectric layers were embedded at the upper and lower surface of the structure. They
implemented the GDQ method to discretize the governing equations. They investigated
the effects of temperature difference, grading index of the material and, etc. Behjat et al.
[15] researched to study the static and dynamic analysis of a functionally graded
piezoelectric plate. They hypothesized that the structure was under the electrical and
mechanical load. They used the first order shear deformation theory and the Hamilton
principle to derive the governing equation. To solve the equations, they implemented the
finite element method. The effects of the materials, boundary conditions and etc were
examined.
Although considerable researches have been devoted to vibrations of double curved
structures, less attention has been paid to vibrations of smart double curved structures in a
thermal condition. In the present paper, the considered structures are composed of three
main layers. They consist of a core made of functionally graded carbon nanotube (FG-
CNT), and two piezoelectric layers in the upper and the lower surfaces. The rule of
mixture approximates the mechanical properties of the FGM. Also, the electrical field
was approximated as a linear function. The main focus of this study is on the four types
of elliptical, spherical, cycloidal and toro circular shells and panels. These structures are
assumed to be moderate thick shells, and the first-order shear deformation theory is also
implemented. The developed model is then solved using both 1D and 2D GDQ [14]. It
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should be noted that the 2D method is complicated in comparison to the 1D mode. Based
on the obtained numerical solution using GDQ methods, a parametric sensitivity analysis
is presented to show the effect of different parameters including mechanical and electrical
boundary conditions, volume distribution, and thermal environment on the vibrations of
the considered structures.
2. Mathematical modeling
The geometric schematics and coordinate systems are exhibited in Fig. (1)-(4). The and
values correspond to the meridional and circumferential directions coordinates. The
position of any point in the shells is determined by 0 1 , 0 1 and
2 2
h hz . The relation of structures’ radii are defined as below [16]. In fig. 2-4 , oz is
known as the axis of revolution and o z is the geometric axis of meridian. Also, the
semi- major and semi- minor axis of the eliptical shell is represented by a and b
respectively. The distance between oz and o z is shown with Rb . The two radiouses of
the structure are displayed with Rα , Rβ.
a) Elliptical shell
2 2
2 2 2 2 3
2
2 2 2 2
(1 )( sin ( ) cos ( ))
sin( )( sin ( ) cos ( ))
b
a bR a
a b
RaR
a b
0 2
(1 )
sin( ) (1 )
b
R R c
b) Cicloidal shell
00 (2 sin(2 )) (2a) 4 cos( ) (2b) (2c)
sin( )c c
RR r R r R
8
c) Circular shell
0
0 (3a) sin( ) (3b) R = (3c) sin( )
b
RR R R R R
In this research, based on the first order shear deformation theory,
The displacement fields are assumed as below [17].
( ( , , ) (4 )
( , , , ) ( , , )
, , , )
, ,
, ,
U z t z t
z
u t a
V z tt v t
(4 )
( , , , ) ( , , ) (4 )
b
z t w cW t
In the above equations u, v and w are displacements of the reference plane along the meridional,
circumferential and normal directions, respectively. Also, , are rotations around the ,
axes, respectively. It should be noted that the above equations ,U V change linearly through the
thickness and W remains constant. Therefore, this assumption can be used for the small
deflection theory. The general form of the strain-displacement relationships is represented as
follows [17].
3
1
2 2
1( ) (5 )
1[ ( ) ( )] i
i k ii
ki i i k k
jiij i j
i j j i i j
u u Aa
A A A
uuA A j
A A A A
(5 ) b
If the above equations are extended, the relationships are developed as below [17].
1 11 3
1 1 2 2 1
2 22 23
2 2 1 1 2 2
1 1( ) (6 ) ( ) (6d)
1 1 1( ) (6 ) (
a aU WV w a
A a R z
a aVU w b
A a R A
2
2 2
2 112 13 1
1 1 2 2 2 1 1 1 1
) ( ) (6 )
1( ) ( ) (6 ) ( ) ( ) (6 )
W VA e
z A
A AV U W Uc A f
A A A A A z A
9
In the above equations , ,U V W are displacements that are substituted by (4a-c) and 1 2 3, ,A A A are
constants which are defined by the following equations[18].
1 1 2 2 3 3
1 2
(1 ) (7a) (1 ) (7b) 1 (7c)z z
A a A a A aR R
The ratio of thickness to radius is zero because it is assumed the shells moderately thick. With
this assumption, the following parameters are defined [18].
Plate:
1 1 (8a) A2=1 (8b) A
Cylindrical shell:
1 21 (9a) (9b) A A R
Double curved shell:
1 2 (10a) sin( ) (10b) A R A R
By substitution (4a-c) into (6a-f) the relationship between strains and mid-surface displacements
are represented as below [17]:
1 2
1 1 2 2 1 2
1 1 (11 ) (11 )
A Au v w v u wa b
A A A R A A A R
2 1 1
1 2 2 1 1 1 2
1( ) ( ) (11 ) (11 )
A A Av uc d
A A A A A A A
2 2 1
2 1 2 1 2 2 1
1 (11 ) ( ) ( ) (11 )
A A Ae f
A A A A A A A
,
1 2
1 1 (11 ) (11 )z z
u w v wg h
R A R A
10
In the above equations,,, are strains in the meridional, circumferential direction and
shearing elements, respectively, , , are the analogous curvature alterations. The material
of the shells vary through the thickness direction based on the FGM model, and their strain
relationships are linear. Therefore the constitutive equations are shown as below [19]. It should
be noted that because of the piezoelectric properties and the presence of the thermal
environment, resultant forces are developed as below:
11 12 11 12
12 22 12 22
66 66
11 12 11 12
12 22 12 22
66 66
55
44
0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0z
N A A B B
N A A B B
N A B
M B B D D
M B B D D
B DM
AQ
AQ
15
24
31 32
0 0
(12 )
0 0
0
0
0 0 0 0
0 0 0 0
0 0 0
p T
p T
p T
p T
p
p
z
z
zz
N N
N N
M Ma
M M
Q
Q
D e
D e
D e e
11
22
33
0 0
0 0 (12 )
0 0 z z
k E p
k E T p b
k E p
In the above formula, is the factor of shear correction being assumed as 5/6. , ,N N N are
force resultants in the meridional, circumferential and in-plain shearing parameter, respectively.
, ,M M M are the moment resultants, and ,Q Q
are the transverse shear force outcomes.
Piezoelectric and thermal resultants are displayed by p and T indices, respectively. , ,D D D
are electrical displacements and ijk is dielectric permittivity.
ijp is pyroelectric property,ije is the
piezoelectric stress constant and iE is the components of the electrical field. The extensional
stiffnessijA , the bending stiffness’s
ijD and the bending-extensional coupling stiffnesses ijB are
defined as below:
11
2 2 22 2 2
2 2 2
( , , ) (1, , ) (1, , ) (1, , ) (13)p
p
h h hh
c p p
h h hh
A B D Q z z dz Q z z dz Q z z dz
In the above relations, index ‘c’ and ‘p’ represent core and piezoelectric parts.
(14a) (14b) (14 )2 2 2 2 2 2
c p p p p
h h h h h hz h z z h c
It is worth noting that piezoelectric layers are substituted in the bottom and top layers.
The components of the electrical field are defined as below [20].
1 2
1 2
1 2 1 2 1( , , ) ( , , ) (15 )
2 2
1 2 1 2 1( , , ) ( , , )
2 2
t tt t t t
z
p p p
b bb b b b
z
p p p
z h z hE E E a
A h A h h
z h z hE E E
A h A h h
(15 )b
In the above equation, is the electric potential that is equal to zero as
1 2 1 2, , ,
Elastic constants of the core and layers are represented as below [21]:
12
11 21 1111 12
12 21 12 21
2222
12 21
(16a) (16 )1 1
(16 ) 1
E EQ Q b
EQ c
66 12
11 12
21 22 11 2
66
(16 )
0
0 (16 ) (16 )1
0 0
C P P
P
Q G d
Q QE
Q Q Q e Q f
Q
Q
12 11 222
66
(16 ) (16 )1
(16 ) 2(1 )
p PP p p
P
pp
p
Eg Q Q h
EQ i
11 12
12 22
66
0
0 (16 )
0 0
p p
p p p
p
Q Q
Q Q Q j
Q
Resultant force and momentum, for the core and layers are represented as [21]:
11 11 112 2 22
22 22 22
2 2
31 31
2
2 2 2
2
( , , ) (1, , ) + +
( , ) (1, ) (1, )
(1
00
6 )
0
p
p
p
p
h hh
p p b t
z z
p ph h hh
T T T c p p p p
h h h
h hh
h
N M F Q z z Tdz Q Tdz Q T
N M z e E dz z e E dz
dz k
2 2
32 32
2 2
15
2
(16 )
( , ) (1, ) (1, ) (16 )
( )
p
p
p
h hh
p p b t
z z
h hh
p
hh
l
N M z e E dz z e E dz m
Q e
2 2
15
2
2 2
24 24
2 2
(16 )
( )
p
p
p
h hh
b t
h
h hh
p b t
h hh
E dz e E dz n
Q e E dz e E dz
(16 )p
13
As mentioned before, the core of the structures are made of FG-CNTs and its mechanical
properties are defined as below [22].
*
*
( ) ( ) (17 )
2( ) 2(1 ) ( )
cnt
cnt
V z V UD a
zV z V FG O
h
*
(17 )
4( ) ( ) (17 )
(
cnt
cnt
b
zV z V FG x c
h
V z
*2( )) (1 ) ( ) (17 )
zV FG V d
h
In the following, the schematic of the CNT dispersion in a matrix displayed as below [23]. In
the above equations, h is the thickness of structures and z indicates the direction of thickness.
The total volume fraction is also exhibited with V*. In the UD (uniformly distributed) type model
(Fig6.a), the distribution of CNT layer in thickness direction remains stable. In the FGV model
(Fig6.b), the higher surface has the maximum distribution of CNT, and the lower surface has the
lowest distribution. FGO(Fig6.c) model has the maximum allocation of CNT in the center of the
section, and it is free of CNT in the top and bottom surfaces. The FGX model( Fig6.d) has the
highest distribution of CNT in the top and bottom surfaces and without CNT in the middle plane.
Also, the mechanical properties of materials are defined as below [22].
211 1 11
22 22
* 2212 12 21 12
11
( ) ( )( ) ( ) (18 ) (18 )
(18c)
cnt m cnt mcnt m cnt m
cnt m
m
V z V zE V z E V z E a b
E E E
EV V
E
3
12 12
11 1111
11
(18 )
( ) ( ) (18 ) ( ) ( ) ( ) (18 )
( ) ( )
( )
cnt mcnt mcnt mcnt m
cnt cnt m m
cnt m
cnt
d
V z V ze z V z V z f
G G G
V z E V z E
V z E
22 12 22 12 11
(18 )( )
(1 ) ( ) (1 ) ( )
cnt m
m
cnt cnt m m
cnt m
gV z E
V z V z
(18 )h
14
In the above equations 1 2 3, , are the first, second and third parameters, respectively which
are all related to CNTs material. 11 22 12, ,cnt cnt cntE E G are the young modulus and shear modulus of
single wall carbon nanotubes (SWCNTs), respectively.cnt and
m are mass densities of the
CNT and matrix constituents, respectively. Also 11 22, are the terms for thermal expansion
coefficients in , directions.
In order to derive the equation of motion of the considered structure, the Hamilton principle is
implemented [24].
0
0 (19)
t
total total totalU V K dt
In the above formula, , ,U V K are virtual strain energy, the virtual potential energy and
virtual kinetic energy, respectively.
1 2
(
)
total z z z zA
z z
U
D E D E D E A A d d
(21)
2 2 2
1 1 1
1 22 2 2
2 2 2
1 1 1( ( ) ( ) ( )
1 1 1( ) ( ) ( )) (22)
T T T
totalA
T T T
U U V V W WV N N N
A A A
U U V V W WN N N A A d d
A A A
1 2(( ) ( ) ( )) (23)totalA
U U V V W WK A A d d
t t t t t t
15
Using the Hamilton principle, the motions of equations are obtained as below:
2 2 12
1 2 1 2 2 1 2 1 1 2 2
2 10 1
1 2 1 1 2 2
1 1 1 1 1( ) ( ) ( )
1 1( ) ( ) (24 )
T T
T T
N QA A Au uA N N N N
A A A A A R A A A A A A
A AM M I u I a
A A A A A A
2 2 12
2 1 2 1 2 1 2 1 1 2 2
2 10 1
1 2 1 1 2 2
1 1 1 1 1( ) ( ) ( )
1 1( ) ( ) (24 )
T T
T T
N QA A Av vN A N N N
A A A A A R A A A A A A
A AM M I v I b
A A A A A A
2 1
2 0
1 2 2 1 2 1 1 2 2
1 1 1 1( ) ( ) ( ) (24 )T T
N QN A Aw wQ A N N I w
R R A A A A A A A A Ac
2 2 12
1 2 1 2 2 1 2 1 1 2 2
2 11 2
1 2 1 1 2 2
1 1 1 1 1( ) ( ) ( )
1 1( ) ( ) (24 )
T T
T T
MA A Au uA M M Q M M
A A A A A A A A A A A
A AF F I u I d
A A A A A A
2 2 12
2 1 2 1 2 1 2 1 1 2 2
2 11 2
1 2 1 1 2 2
1 1 1 1 1( ) ( ) ( )
1 1( ) ( ) (24 )
T T
T T
M A A Av vM A M Q M M
A A A A A A A A A A A
A AF F I v I e
A A A A A A
2 2 2
2 2 2
1 1 2 1 1 2 1 1( ( ) ) ( ( ) ) 0 (24 )
2 2
p p p
h h hh h h
z
h h hp p p
Dz h z hD dz D dz dz f
R R R h R R R h R R R R h
16
2 2 2
2 2 2
1 1 2 1 1 2 1 1( ( ) ) ( ( ) ) 0 (24 )
2 2
p p p
h h hh h h
z
h h hp p p
Dz h z hD dz D dz dz g
R R R h R R R h R R R R h
3. Solution procedure
GDQ method in one-dimensional:
In this work, the GDQ method is applied to solve the equations of motion of the considered
structures. In order to solve the equations, at first, we assume that they are constant in the
circumferential direction. As a result, Eqs. 25(a-g) are used to decrease the 2D equations to 1D
[25].
( , , ) ( , )cos( ) (25 ) ( , , ) ( , )sin( ) (25 )
( , , ) ( , )cos( ) (25 ) ( , , ) ( , )cos( )
u t u t n a v t v t n b
w t w t n c t t n
(25 )
( , , ) ( , )sin( ) (25 ) ( , , ) ( , )cos( ) (25 )
( , , ) ( , )sin( ) (25 )
d
t t n e t t n f
t t n g
Based on this method, grid points along the reference surface are as follows [26]:
1 00
( )1(1 cos( )) (26)
1 2i
i
N
The final equation of motion is approximated by the 1D GDQ method. Details of this method are
descripted as below [26].
17
1
(27 )n N
n
ik knk
fc f a
1
1,
( ) ( ) (27 )N
k k j
j j k
M b
1
1
1
( ) , 1,2,..... , (27c)
( ) ( )
iij
j j
Mc i j N j i
M
1 ( 1) ( 1)( *( ) / ( )) , 1,2,..., 2,3,.... 1 (27 )n n n
ij ij ii ij i jc n c c c i j j i n N d
1,
, 1,2,..., 1,2,3,.... 1 (27 )N
n n
ii ij
j j i
c c i j N n N e
GDQ method in two-dimensional :
For 2D solution, the approximations are considered in two directions as below [4]:
1
1
(28 )
n Nn
ik kjnk
n Nn
jm imnm
uc u a
uc u
(28 )b
Based on this method, grid points along the reference surface are considered as below:
1 00
1 00
( )1(1 cos( )) (29 )
1 2
( )1(1 cos( ))
1 2
i
j
ia
N
j
M
(29 )b
18
1 1
1, 1,
1
1
1
( ) ( ) (30 ) ( ) ( ) (30 )
( ) , 1, 2,..... ,
( ) ( )
N N
k k j k k j
j j k j j k
iij
j j
M a M b
Mc i j N j i
M
1
1
1
1 ( 1) ( 1)
(30 )
( ) , 1, 2,..... , (30 )
( ) ( )
( *( ) / ( )) , 1, 2,...
iij
j j
n n n
ij ij ii ij i j
c
Mcp i j N j i d
M
c n c c c i j
1 ( 1) ( 1)
1,
, 2,3,.... 1 (30 )
( *( ) / ( )) i,j=1,2,... j i n=2,3,...,N-1 (30f)
, 1, 2,.
n n n
ij ij ii ij i j
Nn n
ii ij
j j i
j i n N e
cp n cp cp cp
c c i j
1,
.., 1, 2,3,.... 1 (30 )
, 1, 2,..., 1, 2,3,.... 1 N
n n
ii ij
j j i
N n N g
cp cp i j N n N
(30 )h
For the panel structures, the boundary conditions are considered as below:
In the panel, for example C-S-C-S is indicative that the panel is clamped at 0 and 1 so
it is simply supported at 0 and 1
Using of the GDQ method, the matrix form of the equations of motion are as below:
[ ] [ ] [ ] 0 (31 )
[ ] [ ] 0
mm mE
Em EE
M m K m K E a
K m K E
(31 )b
In the above equation, it is purposed that
19
(32 )
T
T
b t
m u v w a
E
(32 )b
Indices of b and t indicate the bottom and top layers, respectively. By eliminating of the
electrical variables, the final form of the equation is introduced as [27];
1[ ] ([ ] [ ][ ] [ ]) 0 (33)mm mE EE EmM m K K K K m
For exploring the effect of electrical properties, two boundary conditions are considered
including open and close circuits. The electrical condition is closed, if the upper and lower
surfaces are grounded else it is open.
[ ] ([ ]) 0 (34)mmM m K m
The equation (33) or (34) is separated in two equations.
[ ] (35 )
0
db b dd d d
bb b bd d
K m K m M m a
K m K m
(35 )b
By elimination of bm , the final form is as below:
1
(36)dd db bb bd d dK K K K m M m
In the above equation, the b and d index presents the boundary and inside domain respectively.
20
For 1D solution:
1 1 1 1 1
2 3 1 2 3 1 2 3 1 2 3 1 2 3 1
, , , , , , , , , (37 )
, ,...., , , ,...., , , ,...., , , ,...., , , ,....,
b N N N N N
d N N N N N
m u u v v w w a
m u u u v v v w w w
(37 )b
For 2D solution:
22 23 2 1 32 33 3 1 ( 1)1 ( 1)2 ( 1)( 1), ,..., , , ,... ,..., , ,... (38 )
, , , ,
T
da N N N N N Nm a a a a a a a a a a
a u v w
11 12 1 21 2 ( 1)1 ( 1)
(38 )
, , , , (38 )
, ,... , , ,... , ,
T
d du dv dw d d
ba N N N N N
b
m m m m m m c
m a a a a a a a a
1 2, ,... (38 )
, , , , (38 )
N N NN
T
b bu bv bw b b
a a d
m m m m m m e
For free vibration, i t
d dm m e is assumed. Therefore the final form to provide frequency of
vibration is as below:
1 2( [ ] ) 0 (39)dd db bb bd dK K K K M m
4. Numerical analysis and discussion
In the current study, the free vibration responses of the FG-CNT double- curved shells and
panels which are embedded with piezoelectric layers are studied. In the first stage, results are
21
compared to other studies. Polymethyl methacrylate(PMMa) is chosen as a matrix with the
following material properties [28].
mE = (3.52−0.003* T ) GPa, m = 0.34 and m = 1150 kg/m3. Here, T = T −T0. The
mechanical properties of SWCNT at different temperatures are given in Table 3.
For validation, results are initially obtained from shells and panels via the GDQ method, and
then compared with other works. For the investigation of validity, the materials are purposed as
FGM model and made of zirconia (ceramic) and aluminum (metal). Young modulus, Poisson
ratio and mass density for Zirconia are assumed as below:
cE = 168GPa, c = 0.3, c = 5700 kg/m3, and for Aluminum mE = 70Gpa , m =0.3, m = 2707
kg/m3, respectively. The volume fraction of FGM is considered as
1
1 1( / / / ) : [1 ( ) ( ) ]
2 2
c pz zFGM a b c p V a b
h h
2
1 1( / / / ) : [1 ( ) ( ) ]
2 2
c pz zFGM a b c p V a b
h h . It is worth noting that for validation, the
spherical shell is considered, and the results are compared to [29].
The geometrical properties of the spherical shell are assumed as below:
0 1, , 0.05, 18 2
h R
.
It should be noted that these results are obtained without piezoelectric materials and at the room
temperature. For validation of the panel’s analysis, the toro circular panel is analyzed. Then, the
results were compared with [30]. The obtained results are explained in Table 7. For this purpose
the materials are assumed as FGM and the volume fraction is as below:
1( )2
pzFGM
h and
1( )2
pzFGM
h .
Also,geometrical properties of this structure are as below:
0 1
2 23, 0.2, , , , 1.5
3 3 3bR h R
,
3
212(1 )
mm
m
E hD
.
22
To investigation of the accuracy for the structures with piezoelectric materials, the results are
obtained for a square plate. The plate is made of FG-CNT and embedded with piezoelectric in the
top and bottom surfaces. The results are reported in Table 8. The piezoelectric being considered
in this study is PZT-5A. It is worth noting that the parameters in the plate structure are
1 21, 1, ,A A R R . Subsequently, by substituting these parameters in the governing
equations, the results are produced for the plate. Table 3-5 report the material properties of the
CNT. Also, the geometrical properties of the plate are as below:
1 10.4, 0.4, 0.05, 0.1x y h hp h
4.1. Case study result
In this section, the effects of electrical and mechanical boundary conditions, the distribution of
CNTs, thermal environment, the volume fraction of CNTs and the geometric characteristics of
the shells and panels are studied. Tables 9-12 present the fundamental frequency of the spherical,
toro circular, cycloidal and elliptical shells that are reinforced with CNTs and integrated with
piezoelectric layers. The natural frequencies of the four structures with three boundary
conditions including C-C, C-S, and S-S are represented in Tables 9-12. Based on these results,
the switching of the electrical boundary conditions (change from closed-circuit to the opened-
circuit) leads to an increase in the amount of the fundamental frequencies. With comparing the
results of the CNTs pattern distributions, we come up to this understanding that the FG-X type of
CNTs distribution has the highest frequency, while the FG-O pattern has the lowest frequency
value. Also, it is noticed that between three cases of the boundary conditions, C-C shells have
higher natural frequencies in comparison with C-S and S-S boundary condition. Furthermore,
Tables 13-16 show that by changing the electrical state to opened-circuit, the fundamental
frequencies of the panels increase. All of the case studies show that with increasing of the CNT
volume fraction, the fundamental frequencies increase. Furthermore, between four possible
graded patterns of the CNTs, the FG-X type of CNT distribution results in higher frequencies,
and the lowest frequencies are relevant to the FG-O pattern. As it was expected, among the two
boundary conditions, CCCC panels have higher natural frequencies in comparison to the CSCS
panels. Figs. 7-14 show the fundamental frequencies of structures in a thermal environment. For
investigation of the thermal effect, the distribution function of CNT is asssumed based on FGO
23
pattern, and the volume fraction is considered as 0.12. According to these figures, the
fundamental frequencies decrease with increasing the temperature value. So by increasing the
ratio of the piezoelectric layer thickness to the core thickness, the effect of temperature is
negligible and can be ignored. Thus, the difference between fundamental frequencies at different
temperatures decreases.
5. Conclusion
In this study, the GDQ method is implemented to examine the free vibration of double curved
structures. The core of the structures is made of CNTs. It is noticed that the mechanical
properties are obtained based on the modified of the rule of mixture. The upper and lower
surfaces are piezoelectric layers. The numerical results are employed to examine the effects of
the boundary conditions, volume fractions, electrical condition, the thickness of piezoelectric
layers and kinds of structures. It is observed that the frequency parameter of the whole structures
rises when the FGX pattern is used. Also, when the volume fraction is increased, this parameter
raises. Plus, the opened-circuit condition has a higher frequency than the closed- circuit one. In
the high mode number, the difference between opened-circuit and closed-circuit frequencies
increase. The influence of the mechanical boundary conditions on the fundamental frequencies is
studied; the clamped-clamped boundary condition has a higher frequency than others. Also, the
effect of a thermal environment is investigated. In as much as by increasing the environment
temperature, the frequency parameters tend to decrease. By increasing the piezoelectric layers
thickness, the influence of a thermal environment decreases.
Funding This article has received no funding.
24
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27
Table captions:
Table1: Bounday conditions that considered for the shell structures.
Table 2: Bounday conditions that are considered for the panel structures.
Table 3: Thermo mechanical properties of (10.10) armchair SWCNT at some specific
temperatures.
Table 4: The parameters of (10, 10) armchair SWCNT obtained from molecular dynamics.
Table 5: Mechanical and electrical properties of PZT-5A.
Table 6: Comparison of frequencies (HZ) for the FGM spherical shell with different boundary
conditions. Geometrical properties of the spherical shell are :
0 1, , 0.05, 18 2
h R
.
Table 7: Comparison of frequency parameters 2 m
m
hR
D
for FGM Toro circular
panel.Geometrical properties of this structure are : 0 1
2 23, 0.2, , , , 1.5
3 3 3bR h R
Table 8: Comparison of frequency parameters
2
1 m
m
x
h E
for CCCC plate integrated with PZT-5A
layers. The geometrical properties of the plate are : 1 10.4, 0.4, 0.05, 0.1x y h hp h
Table 9: Fundamental frequencies of the spherical shell . Geometrical parameters are : 0 1,
8 2
(,h=0.05m, R=1m, hp=0.1h, n=1).
Table 10: Fundamental frequency of Toro circular shell. Geometrical parameters are: ( 0 1
2,
3 3
,h=0.2m, hp=0.1h , Rb=1.5m, R=3m).
Table 11: Fundamental frequency of Elliptical shell. Geometrical parameters are: ( 0 1
5,
6 6
,
h=0.2m, a=3m, b=2m, Rb=0, hp=0.1h, n=1).
Table 12: Fundamental frequency of Elliptical shell. Geometrical parameters are: ( 0 1
5,
6 6
, h=0.2m, a=3m, b=2m, Rb=0, hp=0.1h, n=1).
28
Table 13: Fundamental frequency of Spherical panel ( 0 1
2, ,
8 2 3
, h=0.05m, R =
1m, hp=0.1h).
Table 14: Fundamental frequency of Toro circular panel. Geometrical parameters are: (
0 1
2 2, ,
3 3 3
, h=0.2m, R = 3m, Rb=1.5m, hp=0.1h).
Table 15: Fundamental frequency of Cycloidal panel. Geometrical parameters are : (
0 1
5 2, ,
12 12 3
,h=0.1m, Rb=2m , rc=1m , hp=0.1h).
Table 16: Fundamental frequency of Elliptical panel. Geometrical parameters are : (
0 1
5 2, ,
6 6 3
, h=0.2m, Rb=0 , hp=0.1h ,a=3,b=2).
29
Figure captions:
Fig1: Co-ordinate system of doubly curved shell.
Fig 2: Co-ordinate system of elliptical shell.
Fig3: Co-ordinate system of cycloidal shell.
Fig4: Co-ordinate system of toro circular shell.
Fig 5: a)Toro circular shell, a’) cross section of toro circular shell, b)spherical shell ,
b’)spherical panel ,c)Elliptical shell, c
’)Elliptical panel, d)cycloidal shell, and d
’) cycloidal panel.
Fig6: Shematic of CNT dispersion in structures a) UD, b) FGV, c) FGO d) FGX.
Fig7: Fundamental frequency of spherical shell with ratio of thickness of piezoelectric layer to
core layer (c-s).
Fig8: Fundamental frequency of toro circular shell with ratio of thickness of piezoelectric layer
to core layer (c-s).
Fig9: Fundamental frequency of cycloidal shell with ratio of thickness of piezoelectric layer to
core layer (c-s).
Fig10: Fundamental frequency of elliptical shell with ratio of thickness of piezoelectric layer to
core layer (c-s).
Fig11: Fundamental frequency of spherical panel with ratio of thickness of piezoelectric layer to
core layer (c-s).
Fig12: Fundamental frequency of toro circular panel with ratio of thickness of piezoelectric layer
to core layer (c-s).
Fig13: Fundamental frequency of cycloidal panel with ratio of thickness of piezoelectric layer to
core layer (c-s).
Fig14: Fundamental frequency of elliptical panel with ratio of thickness of piezoelectric layer to
core layer (c-s).
30
Table1.Bounday conditions that considered for the shell structures.
C-C i=1 , i=N 0i i i i iu v w M
0i i i i iu v w
C-S
i=1
i=N 0i i i i iu v w
0i i i i iu v w M
S-S i=1 , i=N 0i i i i iu v w M
Table2.Bounday conditions that are considered for the panel structures.
C-C-C-C i=1,2,…M
j=1,2,..,N 0ij ij ij ij iju v w
C-S-C-S
i=1,M , j=1,2,…N
i=1,2,..M , j=1,N
0ij ij ij ij iju v w M
0ij ij ij ij iju v w M
Table3
Thermo mechanical properties of (10.10) armchair SWCNT at some specific temperatures [31].
𝜌𝑘𝑔
𝑚3 𝜈12
𝐶𝑁𝑇 𝐺12𝐶𝑁𝑇(𝑇𝑃𝑎)
𝛼11𝐶𝑁𝑇 ∗ 10−6(
1
𝑐) 𝐸22
𝐶𝑁𝑇(𝑇𝑃𝑎) 𝛼22𝐶𝑁𝑇 ∗ 10−6(
1
𝑐) 𝐸11
𝐶𝑁𝑇 T
1400
1400
1400
0.17
0.17
0.17
1.94
1.96
1.96
3.45
4.53
4.53
7.08
6.93
6.86
5.16
5.01
5.01
5.64
5.53
5.47
300
500
700
31
Table 4
The parameters of (10, 10) armchair SWCNT obtained from molecular dynamics [32].
3 2 1 *
CNTV
0.12 0.137 1.022 0.7𝜂2
0.17 0.142 1.626 0.7𝜂2
0.18 0.141 1.585 0.7𝜂2
It is worth noting that, the shear modulus 13G is taken equal to 12G whereas 23G is taken equal to
121.2G [32].
Table 5
Mechanical and electrical properties of PZT-5A [33].
E(Gpa) ν ρ (𝐾𝑔
𝑚3) Electro mechanical (𝑐
𝑚3) Permittivity coefficients
( F/m×10-8
)
63
0.35
7600
𝑒31 = −7.209
𝑒32 = −7.209
𝑒15 = 12.322
𝑒24=12.322
K11 = 1.53
K22 = 1.53
K33 = 1.5
32
Table 6
Comparison of frequencies (HZ) for the FGM spherical shell with different boundary conditions.
Geometrical properties of the spherical shell are :
0 1, , 0.05, 18 2
h R
.
[29]
C C
present reference
[29].
S S
present reference
n P
910.55 910.69
894.65 894.68
889.43 889.45
827.82 827.89
886.88 886.90
881.63 881.64
1
2
3
FGMI (a=1/b=0/c/p=0.6 )
887.00 887.13
859.35 859.37
856.54 856.57
797.35 797.41
851.91 851.92
848.52 848.53
1
2
3
P=20
908.66 908.80
893.78 893.81
888.30 888.32
815.15 815.21
887.92 887.95
885.54 885.56
1
2
3
FGMII (a=1/b=0/c / p =0.6)
885.80 885.93
858.73 858.76
855.74 855.76
790.75 790.81
852.66 852.68
850.57 850.59
1
2
3
P=20
33
Table 7
Comparison of frequency parameters 2 m
m
hR
D
for FGM Toro circular panel.Geometrical
properties of this structure are :
0 1
2 23, 0.2, , , , 1.5
3 3 3bR h R
[30]
CSCS
persent reference
[30]
CCCC
present reference p Type
60.38 59.47 63.06 63.05 0.5
3
FGMI
59.06 58.08 61.41 60.48 10
57.53 56.66 57.77 56.90 0.5
3
2
56.01 55.16 56.30 55.45 10
56.74 56.06 56.79 56.07 0.5
3
4
55.23 54.52 55.26 54.53 10
60.71 59.79 62.95 62.00 0.5
3
FGMII
59.33 58.43 61.30 60.37 10
57.51 56.65 57.67 56.82 0.5
3
2
55.96 55.13 56.18 55.34 10
56.65 55.99 56.69 55.99 0.5
3
4
55.12 54.43 55.15 54.44 10
34
Table 8
Comparison of frequency parameters
2
1 m
m
x
h E
for CCCC plate integrated with PZT-5A
layers. The geometrical properties of the plate are : 1 10.4, 0.4, 0.05, 0.1x y h hp h
[20]persent reference
condition Type *
CNTV
5.46 5.32 open FGO 0.12
5.23 5.19 close
5.75 5.60 open FGO 0.17
5.52 5.47 close
6.12 6.01 open FGO 0.28
5.93 5.84 close
5.67 5.54 open FGV 0.12
5.45 5.40 close
5.98 5.88 open FGV 0.17
5.80 5.73 close
6.63 6.34 open FGV 0.28
6.28 6.16 close
6.27 6.17 open FGX 0.12
6.09 5.97 close
6.78 6.68 open FGX 0.17
6.62 5.46 close
7.42 7.33 open FGX 0.28
7.29 7.06 close
35
5.91 5.79 open UD 0.12
5.70 5.62 close
6.32 6.21 open UD 0.17
6.14 6.03 close
6.86 6.77 open UD 0.28
6.71 6.54 close
36
Table 9
Fundamental frequencies of the spherical shell . Geometrical parameters are : ( 0 1,8 2
, h=0.05m,
R=1m, hp=0.1h, n=1).
S – S C – S C – C condition Type *
CNTV
682.59 689.76 709.45 open FGX 0.12
682.22 689.31 709.01 close
756.08 766.53 791.89 open FGX 0.17
755.84 766.23 791.60 close
830.13 846.03 880.71 open FGX 0.28
830.03 845.89 880.58 close
650.60 685.59 695.31 open FGV 0.12
650.37 685.26 694.92 close
713.12 757.38 771.04 open FGV 0.17
712.96 757.19 770.79 close
771.87 824.41 846.99 open FGV 0.28
771.79 824.34 846.88 close
657.11 673.19 688.79 open FGO 0.12
656.81 672.78 688.39 close
720.18 741.61 761.90 open FGO 0.17
719.99 741.34 761.65 close
776.16 804.70 833.22 open FGO 0.28
776.07 804.58 833.11 close
37
Table 10
Fundamental frequency of Toro circular shell. Geometrical parameters are: ( 0 1
2,
3 3
,h=0.2m, hp=0.1h ,
Rb=1.5m, R=3m).
S – S C – S C – C condition Type *
CNTV
251.21 252.41 254.63 open FGX 0.12
251.12 252.33 254.57 close
278.91 280.13 282.18 open FGX 0.17
278.85 280.08 282.15 close
316.07 317.17 318.90 open FGX 0.28
316.04 317.15 318.88 close
251.59 251.60 251.64 open FGV 0.12
251.50 251.51 251.55 close
278.42 278.47 278.77 open FGV 0.17
278.35 278.41 278.71 close
313.53 313.76 314.97 open FGV 0.28
313.48 313.72 314.93 close
248.80 249.05 250.13 open FGO 0.12
248.71 248.96 250.05 close
275.96 276.20 277.05 open FGO 0.17
275.90 276.15 277.05 close
312.21 312.36 312.97 open FGO 0.28
312.18 312.33 312.94 close
38
Table 11
Fundamental frequency of Cycloidal shell. Geometrical parameters are : (5
,12 12
, h=0.1m,
hp=0.1h, rc=1m, Rb=2m).
S – S C – S C – C condition Type *
CNTV
214.83 242.99 261.36 open FGX 0.12
214.70 242.88 261.14 close
248.40 278.23 297.95 open FGX 0.17
248.30 278.15 297.79 close
286.92 317.08 340.85 open FGX 0.28
286.86 317.02 340.79 close
190.21 213.09 236.75 open FGV 0.12
190.19 213.04 236.67 close
218.54 242.84 268.04 open FGV 0.17
218.53 242.80 267.98 close
249.91 276.82 302.53 open FGV 0.28
249.90 276.79 302.50 close
186.78 209.49 229.57 open FGO 0.12
186.63 209.35 229.43 close
214.27 238.26 259.30 open FGO 0.17
214.15 238.15 259.20 close
243.96 270.07 291.70 open FGO 0.28
243.88 269.99 291.64 close
39
Table 12
Fundamental frequency of Elliptical shell. Geometrical parameters are: ( 0 1
5,
6 6
, h=0.2m, a=3m,
b=2m, Rb=0, hp=0.1h, n=1).
S – S C – S C – C condition Type *
CNTV
202.51 216.74 231.68 open FGX 0.12
202.46 216.69 231.64 close
224.93 239.11 253.77 open FGX 0.17
224.89 239.08 253.75 close
245.69 258.49 271.42 open FGX 0.28
245.68 258.47 271.41 close
183.11 200.40 219.23 open FGV 0.12
183.09 200.38 219.22 close
202.49 220.52 239.86 open FGV 0.17
202.48 220.51 239.85 close
220.77 238.64 257.19 open FGV 0.28
220.76 238.63 257.18 close
187.53 201.20 216.10 open FGO 0.12
187.47 201.13 216.04 close
206.81 220.95 236.17 open FGO 0.17
206.77 220.91 236.13 close
223.98 238.16 253.11 open FGO 0.28
223.96 238.14 253.09 close
40
Table 13
Fundamental frequency of Spherical panel ( 0 1
2, ,
8 2 3
, h=0.05m, R = 1m, hp=0.1h).
C – S – C – S
C – C – C – C condition Type *
CNTV
784.00 811.12 open FGX 0.12
783.41 810.55 close
871.27 901.86 open FGX 0.17
870.94 901.51 close
958.97 996.07 open FGX 0.28
958.85 995.94 close
758.08 777.66 open FGV 0.12
757.60 777.20 close
831.81 852.34 open FGV 0.17
831.54 852.04 close
897.71 923.00 open FGV 0.28
897.55 922.82 close
737.08 755.56 open FGO 0.12
736.68 755.14 close
800.44 821.97 open FGO 0.17
800.25 821.76 close
857.49 885.10 open
FGO 0.28 857.42 885.03 close
41
Table 14
Fundamental frequency of Toro circular panel. Geometrical parameters are: ( 0 1
2 2, ,
3 3 3
,
h=0.2m, R = 3m, Rb=1.5m, hp=0.1h).
C – S – C – S
C – C – C – C condition Type *
CNTV
282.31 286.34 open FGX 0.12
282.11 286.13 close
320.35 325.13 open FGX 0.17
320.20 324.97 close
374.78 378.00 open FGX 0.28
374.69 377.98 close
273.77 279.06 open FGV 0.12
273.57 278.83 close
309.02 316.16 open FGV 0.17
308.87 315.99 close
349.00 351.02 open FGV 0.28
348.96 350.98 close
273.44 276.56 open FGO 0.12
273.22 276.33 close
309.50 313.27 open FGO 0.17
309.34 313.10 close
336.17 338.65 open FGO 0.28
336.15 338.64 close
42
Table 15
Fundamental frequency of Cycloidal panel. Geometrical parameters are : ( 0 1
5 2, ,
12 12 3
,h=0.1m, Rb=2m , rc=1m , hp=0.1h).
CSCS
CCCC condition Type *
CNTV
260.07 260.68 open FGX 0.12
259.90 260.51 close
294.15 294.51 open FGX 0.17
294.05 294.42 close
328.82 330.13 open FGX 0.28
328.80 330.10 close
235.45 235.61 open FGV 0.12
235.37 235.52 close
262.38 262.57 open FGV 0.17
262.32 262.52 close
289.08 290.50 open FGV 0.28
289.04 290.46 close
225.49 225.62 open FGO 0.12
225.41 225.54 close
249.09 250.32 open FGO 0.17
249.06 250.27 close
272.87 274.39 open FGO 0.28
272.86 274.38 close
43
Table 16
Fundamental frequency of Elliptical panel. Geometrical parameters are : ( 0 1
5 2, ,
6 6 3
,
h=0.2m, Rb=0 , hp=0.1h ,a=3,b=2).
CSCS
CCCC condition Type *
CNTV
257.38 257.83 open FGX 0.12
257.26 257.72 close
278.35 289.32 open FGX 0.17
278.33 289.23 close
295.62 307.81 open FGX 0.28
295.61 307.80 close
250.32 250.54 open FGV 0.12
250.26 250.48 close
265.89 272.96 open FGV 0.17
265.87 272.94 close
279.80 288.04 open FGV 0.28
279.78 288.02 close
239.20 241.96 open FGO 0.12
239.17 241.83 close
255.56 263.69 open FGO 0.17
255.54 263.67 close
268.94 278.23 open FGO 0.28
268.93 278.22 close
44
Fig1. Co-ordinate system of doubly curved shell.
Fig 2. Co-ordinate system of elliptical shell.
45
Fig3. Co-ordinate system of cycloidal shell.
Fig4. Co-ordinate system of toro circular shell.
46
Fig 5. a)Toro circular shell, a’) cross section of toro circular shell, b)spherical shell , b
’)spherical panel ,c)Elliptical
shell, c’)Elliptical panel, d)cycloidal shell, and d
’) cycloidal panel.
Fig6. Shematic of CNT dispersion in structures a) UD, b) FGV, c) FGO d) FGX.
47
Fig7.Fundamental frequency of spherical shell with ratio of thickness of piezoelectric layer to core layer (c-s).
Fig8.Fundamental frequency of toro circular shell with ratio of thickness of piezoelectric layer to core layer (c-s).
48
Fig9.Fundamental frequency of cycloidal shell with ratio of thickness of piezoelectric layer to core layer (c-s).
Fig10.Fundamental frequency of elliptical shell with ratio of thickness of piezoelectric layer to core layer (c-s).
49
Fig11.Fundamental frequency of spherical panel with ratio of thickness of piezoelectric layer to core layer (c-s).
Fig12.Fundamental frequency of toro circular panel with ratio of thickness of piezoelectric layer to core layer (c-s).
50
Fig13.Fundamental frequency of cycloidal panel with ratio of thickness of piezoelectric layer to core layer (c-s).
Fig14.Fundamental frequency of elliptical panel with ratio of thickness of piezoelectric layer to core layer (c-s).
51
Biographies:
Siros Khorshidi graduated in Mechanical Engineering from Tehran Polytechnic in 2016. He
also received his BSc in material science and engineering from the University of Tehran in 2012.
His current research interests include nonlinear vibration.
Saeed Saber Samandari is currently an Assistant Professor in New Technologies Research
Center of the Amirkabir University of Technology. He has researched experimental, analytical,
and computational aspects of the properties of engineered materials, particularly at the sub-
microscopic and nanoscopic length scales.
Manouchehr Salehi obtained his Ph.D. in Mechanical Engineering from Lancaster University.
He is currently a Professor at the Department of Mechanical Engineering at the Amirkabir
University of Technology. His current research contains Composites, Plates and Shells,
Elasticity, Viscoelasticity, Nanomechanics.