1-s2.0-S0263822316311072-main

Accepted Manuscript
Active vibration control of FGM plates with piezoelectric layers based on Reddy’s higher-order shear deformation theory
B.A. Selim, L.W. Zhang, K.M. Liew
PII:
DOI:
Reference:

S0263-8223(16)31107-2
/>COST 7659

To appear in:

Composite Structures

Received Date:
Accepted Date:

3 July 2016
22 July 2016

Please cite this article as: Selim, B.A., Zhang, L.W., Liew, K.M., Active vibration control of FGM plates with
piezoelectric layers based on Reddy’s higher-order shear deformation theory, Composite Structures (2016), doi:
/>
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers
we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and
review of the resulting proof before it is published in its final form. Please note that during the production process
errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.


Active vibration control of FGM plates with piezoelectric layers based on
Reddy’s higher-order shear deformation theory

B.A. Selim1, L.W. Zhang2,*, K.M. Liew1,3,*
1

Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong, China
2

3

College of Information Science and Technology, Shanghai Ocean University, Shanghai 201306, China

City University of Hong Kong Shenzhen Research Institute Building, Shenzhen Hi-Tech Industrial Park, Nanshan
District, Shenzhen, China

Abstract
Literature researching the active vibration control of functionally graded material (FGM) plates
with piezoelectric layers using Reddy’s higher-order shear deformation theory (HSDT) using any
of the element-free methods does not exist. To the best of the authors’ knowledge, this paper is
the first to use Reddy’s HSDT with the element-free IMLS-Ritz method to investigate this
problem. In this study, seven mechanical degrees of freedom (DOF) and one additional electrical
DOF are considered for each node of the discretized domain. The natural frequency results of
two FGM plates with top and bottom piezoelectric layers are compared with the literature in
terms of various electrical and mechanical boundary conditions, volume fraction exponent (n)
and dimension ratios, with obvious agreement. Furthermore, parametric studies are performed,
for the first time, to study the effects of mechanical boundary conditions, n value, FGM plate
thickness-to-width ratio and piezoelectric layer thickness to FGM plate thickness ratio on the
natural frequency increment between open and closed circuit conditions. For the purpose of
active vibration control, a constant velocity feedback approach is utilized. The effectiveness of
two proposed positions, of piezoelectric sensor and actuator layers, to control the vibration of
FGM plates is investigated.

Keywords: Functionally graded material; Piezoelectric materials; Smart structures; Vibration
control; Reddy’s third-order shear deformation theory; Mesh-free method

1


∗

Corresponding authors.
Email address: (L.W. Zhang), (K.M. Liew)

1. Introduction
In the mid-1980s, functionally graded material (FGM) was introduced for the first time
[1]. FGM is characterized by spatial variation in material properties as it is composed of at least
two different components with a gradual changing of volume fraction along at least one direction
[2]. The purpose of this concept is to combine the best properties of two or more constituents.
Additionally, it helps to eliminate the interface problems commonly found in composite
materials in order to achieve a smoother stress distribution [3].
Moreover, smart materials, including piezoelectric materials, have drawn increasing
attention from several researchers due to their applicability to active control of structures. He et
al. [4] utilized the finite element method (FEM) and the classical plate theory (CPT) for the
shape and vibration control of FGM plates with integrated piezoelectric sensors and actuators.
Liew et al. used FEM with CPT [5] and the first-order shear deformation theory (FSDT) [6], [7]
for the active control of FGM plates with integrated piezoelectric sensors and actuators subjected
to a thermal gradient. Ootao and Tanigawa [8] studied a three-dimensional transient
piezothermoelasticity problem for FGM rectangular plate bonded to a piezoelectric layer due to
partial heat supply. Reddy and Cheng [9] provided three dimensional solutions for FGM plates
with piezoelectric actuator layers, utilizing transfer matrix and asymptotic expansion techniques.
Based on the element-free Galerkin method and FSDT, Dai et al. [10] presented an active shape
control and dynamic response suppression model for FGM plates containing distributed

piezoelectric sensors and actuators. Ray and Sachade [11] studied the static analysis of FGM
plates integrated with a layer of piezoelectric fiber reinforced composite (PFRC) material using
FEM and FSDT.
Based on HSDT and von Kármán-type equations, Huang and Shen [12] investigated the
nonlinear vibration and dynamic response of an FGM plate with surface-bonded piezoelectric
layers in thermal environments utilizing an improved perturbation technique. Fakhari and Ohadi
[13] studied the nonlinear vibration control of FGM plates with integrated piezoelectric sensors
and actuator layers under a thermal gradient and transverse mechanical loads using FEM, HSDT
2


and von Kármán-type equations. Fakhari et al. [14] studied the nonlinear free and forced
vibration behavior of FGM plate with piezoelectric layers in a thermal environment under
thermal, electrical and mechanical loads utilizing FEM, HSDT and von Kármán-type equations
too. Based on the differential reproducing kernel (DRK) interpolation, Wu et al. [15] presented a
meshless collocation method for the three-dimensional coupled analysis of simply-supported,
doubly curved FGM piezo-thermo-elastic shells under a thermal load. Farsangi and Saidi [16]
presented an analytical Levy type approach for the free vibration analysis of moderately thick
FGM plates with piezoelectric layers using FSDT. Loja et al. [17] studied the static and free
vibration behavior of FGM plate with piezoelectric skins using B-spline finite strip element
models based on FSDT and HSDT. Using Hamilton's principle, Maxwell's equation and the
Navier method, Rouzegar and Abad [18] presented an analytical solution for the free vibration
analysis of an FGM plate with piezoelectric layers by employing the four-variable refined plate
theory.
Among the available plate theories in the literature, CPT, FSDT and Reddy’s higherorder shear deformation theory (HSDT) are the most popular. CPT is only suitable for thin plates
as it completely neglects the shear effect. However, the shear effect becomes significant as the
plate becomes thicker and ignoring it leads to erroneous outcomes. Although FSDT incorporates
the shear effect, it necessitates the usage of a shear correction factor to satisfy zero shear
conditions on the plate’s top and bottom surfaces, which is a shortcoming. To overcome the
drawbacks of CPT and FSDT, Reddy proposed his HSDT in 1984 [19] which identifies the

displacement field by a third order function in terms of the thickness coordinate. Both CPT and
FSDT are considered to be special cases of HSDT, wherein the shear effect is considered and
there is no necessity to use any shear correction factor [2], [19]. Furthermore, HSDT is valid for
thin, moderately thick and thick plates.
On the other hand, the usage of the moving least-square (MLS) approximation is
considered to represent a turning point, resulting in substantial improvements in the element-free
method. A powerful property of the MLS approximation is that its continuity is related to the
continuity of the weight function. Accordingly, highly continuous approximations can be
obtained by choosing the appropriate weight function [20]. However, MLS approximation still
has some limitations, such as the fact that the resulting system of algebraic equations may be ill3


conditioned. In such cases, there are no mathematical techniques to investigate whether the
system of algebraic equations is ill-conditioned before it is solved. Consequently an accurate
solution for the system of algebraic equations may not be found/correctly found. The improved
moving least-square (IMLS) approximation was proposed to overcome these drawbacks of MLS
in the construction of shape function by utilizing a weighted orthogonal function [21]–[25].
Through reviewing the literature, it is obvious that there exist no studies on the free
vibration and active vibration control of functionally graded material (FGM) plates with
piezoelectric layers using Reddy’s higher-order shear deformation theory (HSDT) with any of
the element-free methods. This paper investigates the free vibration analysis as well as the active
vibration control of FGM plates with piezoelectric layers using Reddy’s HSDT in association
with the element-free IMLS-Ritz method [26]–[31].

2. Functionally graded material (FGM) plates
In this paper, the FGM plate is assumed to have two constituents as it is composed of
metallic and ceramic materials. The effective properties alter throughout the thickness direction,
as follows:


  = 
 −

  +

,

(1)

where
 is the effective material property of the FGM plate (i.e., the modulus of elasticity or
 is the volume fraction of the metallic constituent of the FGM plate which can be expressed
density) and the subscripts m and c denote the metallic and ceramic constituents, respectively.

as:

2 + ℎ
 ,
  = 
2ℎ


(2)

in which ℎ is the height of the FGM plate and n is the volume fraction exponent0 ≤  ≤ ∞.

At any level (z) of the plate thickness, the relationship between the volume fraction constituents
of the metallic and ceramic parts is expressed by:



m +  = 1.

(3)
4

Fig. 1 shows the geometry and coordinate system of a rectangular FGM plate with top and
bottom piezoelectric layers.

3. Theoretical formulation
3.1. Governing equations for Reddy’s HSDT model
Based on Reddy’s HSDT [19], the displacement field can be expressed as:

, ,  =  ,  + ∅ ,  +
!  " #∅ +
' , ,  = ' ,  + ∅( ,  +
!  " #∅( +
%, ,  = % , ,

$%
&,
$
$%
&,
$

(4)

(5)

(6)

where  , ' , %  are displacement components along the , , directions, respectively, of a
point on the plate’s mid-plane z= 0. ∅ and −∅( are the rotations about the  and  axes,

respectively, and the constant
! = − "+ -  in which ℎ. is the total thickness of the plate.
Supposing that / = ∅ +
rewritten as:

012
0

*

,

 and /( = ∅( +

012
0(

 as reported in [32], Eqs. (4)-(6) can be

, ,  =  ,  + ∅ ,  +
!  " / ,
' , ,  = ' ,  + ∅( ,  +
!  " /( ,
%, ,  = % , .

(7)

(8)

(9)

Therefore, the basic mechanical field variables (mechanical degrees of freedom) for each node
based on the current formulation are , ' , % , ∅ , ∅( , / and/( .
The in-plane strains can be expressed as:
5


where

3 = [5 5(( / ( ]7 = 3 + 8! +  " 89 ,

(10)

$/

$∅ ?
$0 ?

<
?
<
<
$
$
;
>
$
;
>
;

>
;
;
$/ >
$∅ >
$'0 >
;
>.
> ,82 =
1 ;
30 = ;
> ,81 = ;;
$
$
$
>
;
>
;
>
;$∅
>
;
>
$
$'
$∅
$/
0>
$/

>
; 0+
>
; +
; +
$ =
: $
$ =
: $
$ =
: $

(11)

@ = [/(A / A ]7 = 3B +  9 8B ,

(12)

However, the out-of-plane shear strains are given by:

where

$%

0?
<∅ +
/
$ >
;
D

3 = ;
,
8
=
3


1
>
/ E.
;∅ + $%0 >
: 
$ =

(13)

The linear constitutive relations are expressed by:

J

N

< 11
;N12
J = ;; 0
HJ L ;
0
G
G
FJ K ;

: 0
IJ M
G  G

N12 0
0
0 ? 5
5 M
N22 0
0
0 >I
G  G
> /
0 N66 0
0 >  ,
H
L
0
0 N44 0 > G/ G
> /
0
0
0 N55 = F  K

(14)

where the material constants are given by:

N!! = !TU SSU , N99 = !TU --U , N!9 = !TU-S
R

S- -S

R

S- -S

U

RSS

S- U-S

, NVV = W!9 , N** = W9" ,NXX = W!" ,

whereY!! and Y99 are the effective Young’s moduli in the principal material coordinate. W!9 ,
W!" and W9" are the shear moduli and '!9 and '9! are Poisson’s ratios.
3.2. Linear constitutive equations of piezoelectric composite plates

6


In the present study, the piezoelectric layers are assumed to be perfectly bonded to the
host FGM plates. The constitutive relationships of the FGM plate with the piezoelectric layers
are given as follows:

Z = [3 − \7 ],

(15)

^ = \3 + _],

(16)

where Z, 3, ] and ^ denote the stress, strain, electric field and electric displacement vectors,
respectively. [, \and _ denote the material constant, the piezoelectric constant and the dielectric
constant matrices, respectively.

The electric field vector E is given as follows:

] = −gradd = −∇d,

(17)

where d is the electric potential difference vector across the piezoelectric layer assuming that
there is an electrical DOF for each node.

The material constant matrix [ is given in the form of:

f g
;
[ = ;] i
;h h
:h h

] h
i h
j h
h fB

h ^B

h
h?
>
h >,
^B >
iB =

(18)

in which

fkl , gkl , ^kl , ]kl , ikl , jkl  = m
f , ^ , i  = m

ℎp ⁄2

ℎp ⁄2

−ℎp /2

n1,,2 ,3 , 4 ,6 o Nkl d,k, l = 1,2,6,

n1, 2 ,4 o Nkl d,k, l = 4,5.

−ℎp /2

\and _ are given as follows:

7

(19)

(20)


0

s\s = t 0

31

0 0 14 15
v11 0
0
0 0 24 25 u ,_ = t 0 v22 0 u.
0
0 v33
32 0 0
0

(21)

3.3. Total energy functional
The plate potential energy is expressed by:

where

1

1
1
1
w = m # Z7 3 − ^7 ]& xy = m # 37 [3 − 37 \7 ] − ]7 _]& xy,
2
2
z 2
z 2

(22)

3 = {3

(23)

8!

89

3B

8B |7 .

The plate kinetic energy is expressed by:
+, ⁄9
1
}= mm
~ 9 + ' 9 + % 9 xxy,
2 z T+,/9

(24)

where ~ is the mass density and  , ' , %  are the components of the mechanical velocity along
the , , directions, respectively.
The external work is given as:
†

€ = m 7 ‚B − d7 ƒB xy + „
z

‡ˆ!

†

 i… − „
7

‡ˆ!

d7 ‰… ,

(25)

where ‚ and iŠ are the external mechanical surface loads and point loads, respectively.
Meanwhile, ƒ and ‰Š are the external surface charges and point charges, respectively.

Hence, the plate total energy function ‹ can be expressed as:

‹ = w − } − €.

(26)

3.4. Two-dimensional IMLS shape function

8


Utilizing the weighted orthogonal basis functions, a detailed formulation of the IMLS
approximation is presented in [33], [34]. A summary of the construction of the IMLS shape
function is illustrated here. Let u(x) be the function of the field variable defined in the domain Ω.
The approximation of u(x) at point x is denoted by uh (x) and this trial function is expressed as:

ℎ Œ = „



Š Œk Œ = Ž ŒŒ,

k=1 k

(27)

where p(x) is a vector of basis functions that commonly consist of monomials of the lowest order
to ensure minimum completeness, m is the number of terms of the monomials and a(x) is a
vector of coefficients given as:

  = n0 , 1 , … ,  o,

(28)

Ž Œ = 1, ,  = 3, ’k“”k,

(29)

Ž Œ = n1, , , , 2 , 2 o = 6, •x“pk
”k.

(30)

which are functions of x. For a two-dimensional problem, the following basis can be chosen:

or

The local approximation at x, as described by Lancaster and Salkauskas [35], is:
– = „
ℎ Œ, Œ



– k Œ = Ž Œ
– Œ,
Š Œ

k=1 k

(31)

– is the point of the local approximation of x. To obtain the local approximation of the
where Œ

function u(x), the difference between the local approximation uh(x) and the function u(x) has to
be minimized using a weighted least-squares method.
Here, we define a function:
™

—=„

˜ˆ!
™

= „

˜ˆ!

%Œ − Œ˜ [+ Œ, Œ˜  − Œ˜ ]9
›

%Œ − Œ˜ [„

‡ˆ!

Š‡ Œ˜  · ‡ Œ − Œ˜ ]9 ,

(32)

where %Œ − Œ˜  is a weight function with a domain of influence, and Œœ (I=1, 2,…, n) are the

nodes with domains of influence which cover the point x.
9

Eq. (32) can be expressed as:

where

— =  − 7 žŒ − ,

(33)

T = 1 , 2 , … ,  ,

(34)

Š! !  Š9 !  … Š› ! 
Š   Š9 9  … Š› 9 
£,
=  ! 9
⋮
⋮
⋱
⋮
Š! ™  Š9 ™  … Š› ™ 
%Œ − Œ! 
0

0
% Œ − Œ9 
žŒ =  
⋮
⋮

0
0

…
…
⋱
…

(35)

0
0
£.
⋮
%  Œ − Œ™ 

(36)

The minimization condition requires:

∂—
= 0,
∂

(37)

fŒŒ = gŒ,

(38)

which leads to the equation system:

where matrices A(x) and B(x) are:

f = Ơ ,

(39)

g = Ư ,

(40)

where u is the vector that collects the nodal parameters of the field variables for all of the nodes
within the support domain.

For ∀ f(x), g(x) ∈ span (p), we define:
10


, © = „



œ=1

%Œ − Œœ Œœ ©Œœ ,

(41)

where , © is an inner product, and span (p) is the Hilbert space.

In Hilbert space span (p), for the set of points {xi} and weight functions {wi}, if functions p1 (x),
p2 (x), , pm (x) satisfy the conditions:
êô , l ¬ = „



k=1

%k Š« Œk Šl Œk  = ­

0 « ≠ ls
«, l = 1,2, … , ,
¯« « = l

(42)

then the function set p1 (x), p2 (x),…, pm (x) is termed a weighted orthogonal function set with a
weight function {wi} about points {xi}. If p1 (x), p2 (x),…, pm (x) are polynomials, then the
function set p1 (x), p2 (x),…, pm (x) is termed a weighted orthogonal polynomials set with the
weight functions {wi} about points {xi}.
From Eq. (41), Eq. (38) can be written as:

Š , Š 

Š , Š  ⋯ Š1 , Š  ? 1 Œ
Š , œ  ?
<
? < 1
Š2 , Š  > ; 2 Œ > ; Š2 , œ  >

.
>;
>=;
⋮ >>
⋮
>; ⋮ > ;
Š , Š = : Œ= :Š , œ =

1 2
< 1 1
; Š2 , Š1  Š2 , Š2  ⋯
;
⋮
⋮
⋱
;
:Š , Š1  Š , Š2  ⋯

(43)

If the basis function set pi(x) ∈ span(p), i = 1, 2, … , m, is a weighted orthogonal function set
about points {xi}, i.e. if

êk ,l ơ = 0,k ≠ l,

then Eq. (43) becomes:
<
;
;
;

:

Š1 , Š1 
0
0
Š2 , Š2 
⋮
⋮
0
0

(44)

⋯
0 ? 1 Œ
Š , œ  ?
<
? < 1
⋯
0 > ; 2 Œ > ; Š2 , œ  >
.
>;
>=;
⋱
⋮
⋮ >>
>; ⋮ > ;
⋯ Š , Š = : Œ= :Š , œ =

Then, the coefficients ai (x) can be directly obtained as follows:

11

(45)


k Œ =
i.e.,

nŠk , œ o
nŠk , Šk o

k = 1,2, … , ,

(46)

– ŒgŒ,
Œ = f

(47)

where
1

1 1
;
РΠ= ; 0
f
;

; ⋮
; 0
:

0
1

nŠ2 ,Š2 o

⋮
0

⋯

⋯

⋱
⋯

0

?
>
0 >
>.
⋮ >
1
>
nŠ,Šo=

(48)

From Eqs. (46) and (31), the expression of the approximation function uh(x) is:

ℎ Œ = ±Œ = „



œ=1

²œ Œœ ,

(49)

where ±Œ is the IMLS shape function:

– ŒgŒ.
±Œ = n²! Œ, ²9 Œ, … , ²™ Œo = Ž¥ Œf

(50)

The aforementioned formulation is the IMLS approximation in which coefficients ai (x) are
obtained directly. Consequently, forming an ill-conditioned or singular equation system is
avoided.
From Eq. (50), we have:

²œ Œ = „



– ŒgŒ] ,
Š Œ[f
lœ

l=1 l

(51)

derivatives of ²œ Œ can be expressed as:

which is the shape function of the IMLS approximation corresponding node I. Then, the partial

12


²œ,k Œ = „



– g + Š f
– g+f
– g  ].
[Šl,k f
k
k lœ
lœ
l

l=1

(52)

The weighted orthogonal basis function set p = (pi) can be formed with the Schmidt method as:

Š1 = 1,…..Šk = “k−1 − „

n“k−1 , Š« o
Š« , k = 2,3, …
«=1 nŠ« , Š« o
k−1

(53)

or can be expressed as:

Š1 = 1,Š2 = “ − 2 ,Šk = “ − k Šk−1 − ”k Šk−2 ,k = 3,4, …
where

k =
”k =

“Šk−1 , Šk−1 
,
Šk−1 , Šk−1 

(54)

(55)

Šk−1 , Šk−1 

,
Šk−2 , Šk−2 

(56)

and “ = ³!9 + 99 or r=x1+x2 for a two-dimensional problem.
Furthermore, using the Schmidt method, the weighted orthogonal basis function set p = (pi) can
be formed from the monomial basis function. For example, for the monomial basis function:
– = Š
–  = 1, 1 , 2 , 12 , 1 , 2 , 22 , … ,
Ž
k

(57)

and the weighted orthogonal basis function set can be formed by:
– −„
Šk = Š
k

– ,Š 
Š
k «
Š , k = 1,2,3, …
«=1 Š« , Š«  «
k−1

(58)

The usage of the weighted orthogonal basis functions in Eqs. (53) and (54) results in fewer

coefficients in the trial function. Thus, the IMLS approximation has better computational
efficiency than the MLS approximation as well as many other shape functions. On the other
hand, the weight function used in Eqs. (32)-(50) plays an important role in the precision of the
13


results. The weight function has nonzero value over only a small neighborhood of xI to generate
a set of sparse discrete equations and has the value of zero outside it. The cubic spline function is
selected as the weight function, and is defined by:

2
I
− 4˜9 + 4˜"
G 3
% − ˜  ≡ %˜  = 4
4
H − 4˜ + 4˜9 − ˜"
3
G3
F
0

where œ =

∥−œ∥
, xxœ
xœ

1
M

2G
,
1
µ“ < |˜ | ⩽ 1L
2
G
µpℎ“%k K
µ“0 ⩽ |˜ | ⩽

(59)

is the size of the support of node I, calculated by:

xœ = x ×
œ ,

(60)

where xmax is a scaling factor and distance
œ is selected by searching for a sufficient number of
nodes to perform interpolation for the field variable. Because the shape function does not have

the Kronecker delta property, the essential boundary conditions cannot be directly imposed. In
this paper, the transformation method reported in [36] is used to enforce the essential boundary
conditions.
3.5. Discrete system equations
For a domain discretized by a set of nodes xI where I = 1,…, NP, displacement
approximation is expressed in the discrete form as follows:

0


ắ '0
'
ẵ
ắ%
%
ẵ 0
ẵ




ẵ

0 = ẵ  Á = „ ²œ Œ œ = „ ²œ  ẵ ,
=1
=1
ẵ
ẵ
ẵ 

ẵ
/
ẵ /
/

ẳ œ À
ℎ
/
¼ À

(61)

whereas ˜ is a displacement nodal parameter associated with node I.
On the other hand, the electric potential variation throughout the piezoelectric layer thickness is
14


assumed to be linear and the electric field E is assumed to have only one component in the Z
direction (Ez), as defined in [37] . The approximation of electric potential difference across the
piezoelectric layer is expressed as:

¿ = ÿ ¿œ ,
ℎ

(62)

where ¿˜ is the electric potential difference across the piezoelectric layer associated with node I

and ÃÄ is the electric potential shape function.

Referring to Eqs. (17) and (62), the electric field E can be written as:

] = −∇ÃÄ ¿˜ = −gÄ ¿˜ ,

(63)

0
0M
g¿ = 1 ,
H L
FℎŠ K

(64)

in which

I

where ℎ… is the thickness of the piezoelectric layer.
By substituting Eqs. (61) and (63) into Eq. (26) and applying the Ritz procedure to the total
energy functional, the following equations of motion are obtained:

ŁỈ + ÇÈÈ  + ÇÈÄ d = i,

where

(65)

Ç¿  − Ç¿¿ d = ‰,

(66)

15


I g M
ắG G
ẵGg1 G
ầ = m ẵ g2
y ẵH L
ẵGg0 G

G
G


ẳFg2 K




f g

I g M
ắG
G
ẵG g1 G
ầ = m ẵ 3 g2
L
y ẵH
ẵG g0 G
G
G


]

h

<
g ^ i h
;

; ] i j h
; h h h f
: h h h ^


h I g MÂ
h ?> Gg”1 GÁ
h > g”2 Á dy,
^ > Hg0 LÁ
G
GÁ
i = Fg2 K
À

I\ M
G\  G


Â
Á
\  Ég¿ ÊÁ dy,
Á
H L
Á
G \ G
F \ K

2
ẳF g2 K

ầ = ầ ,
y

(70)

= m n˘7 ÌËÍ ody,
z

(71)

 = [˜ '˜ %˜ ¿ ˜ ¿(˜ / ˜ /(˜ ]7 ,
$²

œ
<
; $
;
g = ; 0
;
;$

;
: $

(68)

(69)

ầ = m êg _g ơ dy,

where

(67)

0

(72)

0 0 0 0 0?
>

>
$²œ
0 0 0 0 0>,
$
>
>
$²œ
0 0 0 0 0>
$
=

(73)

16


<0 0 0
;

;
g”1 = ;0 0 0
;
;
;0 0 0
:
<0 0
;
;
g”2 =
1 ;0 0
;
;
;0 0
:
<0 0
;
g0 = ;
;0 0
:

$²œ
$
0

$²œ
$

0

0 0?
>

>
$²œ
0 0>,
$
>
>
$²œ
0 0>
$
=

0 0 0

0 0 0

0 0 0

$²œ
$
0

$²œ
$

(74)

0 ?

>

$²œ >
>,
$ >
$²œ >>
$ =

(75)

$²œ
0 ²œ 0 0?
$
>
>,
$²œ
²œ 0 0 0>=
$

(76)

0 0 0 0 0 0 ²œ
E,
g2 = 3
1 D
0 0 0 0 0 ²œ 0

(77)

0

0 0
\ = Ỵ 0
0 0Ï,
31 32 0

(78)

14 15
\ = Ỵ24 25 Ï,
0
0
²

< œ
;0
;0
;
˜ = ; 0
;0
;
0
;
:0

(79)

0 0 0 0 0 0?
²œ 0 0 0 0 0 >
0 ²œ 0 0 0 0 >
>

0 0 ²œ 0 0 0 >,
0 0 0 ²œ 0 0 >
0 0 0 0 ²œ 0 >
>
0 0 0 0 0 ²œ =

(80)

17


œ
< 0
;
; 0
Ì = ; œ!
; 0
;
;
! œ"
: 0

0
œ
0
0
œ!
0

! œ"

0
0
œ
0
0
0
0

œ!
0
0
œ9
0

! œ*
0

0
œ!
0
0
œ9
0

! œ*


! œ"
0

0

! œ*
0

! 9 œV
0

0

! œ" ?
>
0 >
0 >.

! œ* >
>
0 >

! 9 œV =

(81)

As the electric field is assumed to have only one component in the Z direction (Ez), ÇÈÄ in

Eq. (68) can be rewritten as:

7
7
7

ÇÈÄ = m ng›
\› 7 gÄ + gÐ!
\› 7 gÄ +  " gÐ9
\› 7 gÄ ody.
z

(82)

Matricesf, g, ^, ], i, j, fB , ^B and iB can be calculated utilizing both analytical and numerical

require integration over the domain. œ ,œ! ,œ9 ,œ" ,œ* and œV are defined as:

methods, whereas the Gauss integration is utilized in the calculation of other matrices that

œ ,œ! ,œ9 ,œ" ,œ* ,œV  = m

+, /9

~1,,  9 ,  " ,  * ,  V d.

T+, /9

(83)

By substituting Eq. (66) into Eq. (65), the following form can be obtained:
T!
ặ + nầẩẩ + ầẩ ầT!
ầẩ o = i + ÇÈÄ ÇÄÄ ‰.

(84)

3.6. Active vibration control of FGM plates with piezoelectric sensor and actuator layers
In this paper, the velocity feedback control approach is employed for the active vibration
control of the FGM plates. Each FGM plate is assumed to have two piezoelectric layers: one
piezoelectric sensor layer, denoted with the subscript s, and one piezoelectric actuator layer,
denoted with the subscript a. In this paper, we propose two different positions of the
piezoelectric sensor and actuator layers. In the first case (Case I), the actuator is located at the
top of the FGM plate while the sensor is located at the opposite bottom side. Conversely, in the
second case (Case II), both the sensor and actuator layers are located at the same side of the
FGM plate (the upper side). The control processes for these two proposed cases are shown in
18


Fig. 2. In order to couple the input voltage for the actuator layer d and the output voltage from
the sensor layer, the constant gain velocity feedback (W') is utilized as follows:

d = W' d  .

(85)

Assuming that there is no external charge‰, the output voltage from the piezoelectric sensor
layer can be obtained from Eq. (66) as:

d = [Ç−1
¿¿ ] [Ç¿ ] ,

(86)

‰ = [Ç¿ ]  .

(87)

and the sensor charge resulted by deformation from Eq. (66) is expressed as:

The plate will undergo deformation upon applying any external force. Due to the piezoelectric
effect, which is similar to the generator concept, this deformation will result in generating an
electric voltage in the sensor layer. Accordingly, the generated voltage will be amplified and sent
to the actuator layer as input voltage. Here, the converse piezoelectric effect will take place,
which is similar to the motor concept, resulting in the control and suppression of vibration. In
such a case, the magnitude of the actuator layer charge can be expressed by substituting Eqs. (85)
and (86) into Eq. (66) as follows:

‰ = [Ç¿ ]  − W' ĐÇ¿¿ Ị ĨÇ−1
¿¿ ễ ẹầ ề   ,

(88)

ặ + ế + ầẩẩ  = i,

(89)

Õ = WU [ÇÈÄ ]Ư ĐÇT!
ÄÄ Ị [ÇÄÈ ]B .

(90)



By substituting Eqs. (87) and (88) into Eq. (84), the following form can be obtained:

where C is the active damping matrix defined as:
B

4. Numerical results
In this section, comparison studies are presented to prove the accuracy of the current
methodology, which show evident agreement with two models of FGM plates with piezoelectric
19


layers [4], [16]. Furthermore, parametric studies were carried out to study the effects of
mechanical boundary conditions, n value, FGM plate thickness-to-width ratio and piezoelectric
layer thickness to FGM plate thickness ratio on the natural frequency increment between open
and closed circuit conditions. Finally, active vibration control results for FGM plates with two
proposed positions of piezoelectric layers are presented. Two electrical boundary conditions are
considered: closed-circuit and open-circuit. In this paper, a closed-circuit electrical condition
means that both surfaces of the piezoelectric layer are grounded and subsequently the behaviour
of the piezoelectric layer is only structural. However, in an open-circuit condition, the
electromechanical coupling takes place as we impose no restrictions on the electrical potential
difference between the two surfaces of the piezoelectric layer. On the other hand, each of the
four edges of the plate is assumed to have simply supported (S), fully clamped (C), or free (F)
mechanical boundary conditions. A sequence of four letters, containing “S”, “C” and/or “F”, is
employed to indicate these assumptions for the plate’s four edges.
4.1. Comparison studies
The first considered example is a cantilevered FGM plate composed of aluminum oxide
and Ti–6A1–4V materials at room temperature (300 K) as presented by He et al. [4]. Two
G-1195N piezoelectric layers are bonded to the top and bottom surfaces of the FGM plate. As
reported in [4], Young’s moduli, Poisson's ratios and the densities of aluminum oxide and Ti–
6A1–4V materials are 3.2024×1011 N/m2, 1.0570×1011 N/m2, 0.2600, 0.2981, 3750 kg/m3 and
4429 kg/m3, respectively. For the G-1195N piezoelectric layers, the Young’s modulus, Poisson's
ratio and density are 63×109 N/m2, 0.3 and 7600 kg/m3, respectively, and "! , "9 and v"" are

6.1468 C/m2, 6.1468 C/m2 and 15×10−9 F/m, respectively. The plate is a square measuring 0.4 m
in both length and width. Its thickness is 5 mm and the thickness of each piezoelectric layer is
0.1 mm. A conversion study is performed to choose a suitable number of nodes and scaling
factor dmax in order to achieve accurate results for the first natural frequency, as shown in Fig. 3.
A plate with CFFF mechanical boundary conditions and open-circuit configuration at various n
values is used in this conversion study. As a result, the number of 19×19 nodes with dmax=2.1 is
chosen to achieve very good agreement between the present open-circuit results and the available
FEM results reported in [4] for the six lowest frequencies, considering the various mechanical
20


boundary conditions and n values shown in Tables 1 and 2. Owing to the accuracy of the results,
discretization of the plates to 19×19 nodes with dmax = 2.1 is used for all further analyses.
The second example is an FGM plate combined of Al and Al2O3 as the metallic and
ceramic parts, respectively, coupled with two transversely isotropic PZT-4 piezoelectric layers
for various electrical and mechanical boundary conditions, n values and dimension ratios. The
material properties of the Al, Al2O3 and PZT-4 piezoelectric materials are the same as reported in
[16]. Tables 3-8 show detailed comparisons, with evident agreement between the results of the
present model and those of the analytical Levy-type solution provided by Askari Farsangi and
Saidi [16]. Due to the electromechanical coupling, the natural frequencies of the FGM plates
with open-circuit piezoelectric layers are greater than those with closed-circuit electrical
boundary conditions for all mechanical boundary conditions, n values and plate dimensions. This
conclusion is in agreement with that of Askari Farsangi and Saidi [16]. Here, we define the
natural frequency increment (∆) between open and closed circuit conditions as:

=

ỉĩ Tỉíịò
ỉíịò

.

(91)

4.2. Parametric studies
Parametric studies are carried out to show the effects of mechanical boundary conditions,
n value, FGM plate thickness-to-width ratio (hf /a) and piezoelectric layer thickness to FGM
plate thickness ratio (hp /hf) on ∆ for an FGM plate composed of Al and Al2O3 with two PZT-4
piezoelectric layers with the same material properties as the above second example. Table 9
shows the effect of four different mechanical boundary conditions on ∆ for a square plate with hf
/a =0.02 and hp /hf =0.1 (n=0, 1, 15, and 1000). When comparing SSSS, SSSC, SCSC and CCCC
mechanical boundary conditions, the CCCC condition is observed to have the lowest
electromechanical coupling effect as the value of ∆ is less than those of the other three cases.
This observation can be attributed to the lower values of node displacements and rotations due to
the existence of more boundary constraints in the CCCC condition, which results in a lower
electromechanical coupling effect compared to the other mechanical boundary conditions.
Conversely, the SSSS condition is observed to have the highest value of ∆ as the higher values of
21


the displacements and rotations lead to higher values of ∆ due to the piezoelectric effect.
Similarly, the values of ∆ for the SSSC condition are lower than the ∆ of SSSS and greater than
the ∆ of SCSC.
It is noteworthy that it is only for the second example reported in [16] where n=0 means
that the ceramic part will be dominant as the FGM plate becomes an isotropic Al2O3 plate.
Conversely, n=∞ means that the metallic part will be dominant as the FGM plate becomes an
isotropic Al plate. In this study, as the Young’s modulus of the ceramic part is greater than that
of the metallic part, the lower values of n (the ceramic part is more dominant) are observed to
have higher values of natural frequencies owing to the higher stiffness of Al2O3 compared to Al.

However, this higher stiffness will lead to lower values of node displacements and rotations
resulting in lower values of ∆. For a square plate with hf /a =0.02 and hp /hf =0.1, Fig. 4 shows the
effect of n value on ∆ for the first mode and also confirms the results of Table 9 regarding the
effect of mechanical boundary conditions on ∆. For a square plate with hp /hf =0.1 (n=0, 1, 15,
and 1000), the effect of FGM plate thickness-to-width ratio (hf /a) on ∆ for the first mode is
studied, as shown in Fig. 5(a-b). It is observed that, as the FGM plate thickness-to-width ratio
increases, the value of ∆ is slightly decreased. In contrast, for a square plate with hf /a =0.02
(n=0, 1, 15, and 1000), the effect of piezoelectric layer thickness to FGM plate thickness ratio (hp
/hf) on ∆ is more sensible as the increase in the piezoelectric layer thickness leads to greater
electromechanical coupling effects and a higher value of ∆, as shown in Fig. 5(c-d).
4.3. Active vibration control studies
Two cases of piezoelectric layer positions are studied to investigate the effectiveness of
the current active vibration control approach, as shown in Fig. 2. The material properties of the
FGM plate and the piezoelectric sensor and actuator layers are chosen so as to be similar to those
of the aforementioned first example in which the FGM plate is composed of aluminum oxide and
Ti–6A1–4V materials with two G-1195N piezoelectric sensors and actuator layers. Both the
length and width of a cantilevered (CFFF) FGM square plate and the sensor and actuator layers
are set at 0.3 m. The FGM plate thickness is assumed to be 5 mm and the thickness of each

sensor and actuator piezoelectric layer is set at 0.1 mm. The plate is initially set under a •0 =
100 N/m2 uniformly distributed load, downwards in the vertical direction. Subsequently, the
22


load was removed, resulting in the generation of motion from the initial displacements. The tip
deflection response of the plate is calculated for a time step of 0.001 s and the system is solved as
a state-space model.
4.3.1. Case I: the actuator and sensor are located at two opposite sides
In this case, the actuator is located at the top of the FGM plate while the sensor is located
at the opposite bottom side, as shown in Fig. 2(a). It is noteworthy that the sensor and actuator

layers in the velocity feedback control approach need to be collocated to guarantee that the
damping matrix is always positive definite. Otherwise, any negative damping will result in
vibration amplification instead of vibration suppression. When the value of the volume fraction
exponent goes to zero (n=0) or infinity (n=∞), the FGM plate is considered to be an isotropic
material with zero in-plane displacements at the mid-plane, as the stretching-bending coupling

effect does not exist and the term g7› \› 7 gÄ  in Eq. (82) can be neglected. In order to achieve

direction of the applied voltage while the sensor layer is polarized in the opposite direction as á

a symmetric positive definite damping matrix in such a case, the actuator layer is polarized in the
and ፠have different signs, as shown in Fig. 2(a). In this way, the active damping matrix
expressed in Eq. (90) will be a symmetric positive definite matrix and the system will be stable.
Figs. 6 and 7 show the tip deflection responses for n=0 and ∞, respectively.
It is clear that, as the feedback control gain increases, the decay of the response is observed
to be faster. For n values in-between 0 and ∞, the system becomes unstable due to the effect of
the stretching-bending coupling. This conclusion is in agreement with that of Wang et al. [37]
who reported the same problem in system stability for antisymmetric angle-ply ([-45°/45°/45°/45°]) and attributed it to the stretching-bending coupling effect. To overcome this problem,
the second case of piezoelectric layer positions is proposed in the next section.
4.3.2. Case II: the actuator and sensor are located at the same side
In this case, the sensor and actuator are both located at the same side of the FGM plate, as
case, we do not need to set the sensor layer with opposite polarization direction as á and á

shown in Fig. 2(b). In order to achieve a symmetric positive definite damping matrix in such a

have the same signs, as indicated in Fig. 2(b). However, another problem arises for the
23


calculating of the volume fraction according to Eq. (2) as the mid-plane of the FGM plate with

thickness ℎâ  does not coincide with the mid-plane of the total composite plate with
thicknessℎ. , as shown in Fig. 2(b). In such a case, Eq. (2) can be rewritten as:

2 + x + ℎ
 ,
2ℎ

  = 



(92)

in which d is the difference between the mid-plane of the total plate thickness ℎ.  and the FGM

thickness nℎ o, as indicated in Fig. 2(b). For various n values including 0 and ∞, this proposed
location for the sensor and actuator layers shows stable results for tip deflection responses, as
shown in Figs. 8-11.

5. Conclusions
In this paper, a novel and effective approach based on Reddy’s HSDT in association with
the element-free IMLS-Ritz method is presented to investigate the free vibration and active
vibration control of FGM plates with piezoelectric layers. The free vibration results were
compared with the literature for various configurations, with evident agreement. Novel
parametric studies were carried out to investigate the effects of mechanical boundary conditions,
n value, FGM plate thickness-to-width ratio and piezoelectric layer thickness to FGM plate
thickness ratio on the natural frequency increment between open and closed circuit conditions
(∆). A constant velocity feedback method is used for the active vibration control of FGM plates
with piezoelectric layers considering two proposed positions of sensor and actuator layers. It is

found that: (i) the natural frequencies of the FGM plates with open-circuit piezoelectric layers are
always greater than those with closed-circuit electrical boundary conditions because of the
electromechanical coupling effect; (ii) greater boundary constraints result in higher values of
natural frequencies and lower values of ∆; (iii) plates with higher stiffness always have lower
values of node displacement and rotations resulting in lower values of ∆; (iv) an increase in the
FGM plate thickness-to-width ratio (hf /a) results in a slight decrease in the value of ∆; (v) an
increase in the piezoelectric layer thickness to the FGM plate thickness ratio (hp /hf) results in a
greater electromechanical coupling effect and a higher value of ∆; (vi) the placement of sensor
and actuator piezoelectric layers on two opposite sides is effective for the active vibration control
24