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:shahruz@aut.ac.ir

2

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

3

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

4

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

5

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.

6

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

7

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.