# Integrated and Consistent Active Control Formulation and Piezotransducer Position Optimization of Plate Structures considering Spillover Effects

Authors: Mojtaba Biglar; Hamid Reza Mirdamadi
Journal: Shock and Vibration
 (2014)

---

## Abstract

This study addresses new formulation for active vibration control of plates by optimal locations of attached piezotransducers. Free vibrations are solved by Rayleigh-Ritz and transient by assumed modes methods. Optimal orientations of patches are determined by spatial controllability/observability, as well as residual modes to reduce spillover. These criteria are used to achieve optimal fitness function defined for genetic algorithm to find optimal locations. To control vibrations, negative velocity feedback control is designed. Results indicate that, by locating piezopatches at optimal positions, depreciation rate increases and amplitudes of vibrations reduce effectively. The effect of number of piezodevices is analyzed.


---

## Body

## 1. Introduction

In recent years, the active vibration control has been an important challenge for space structures. The importance has been more remarkable and higher for larger and more flexible space structures. For vibration suppression, smart materials such as piezoelectric transducers may be used [1, 2]. Therefore, determining smart actuator and sensor locations is a key subject for increasing the system efficiency.Many control algorithms have been used to eliminate the vibrations, such as positive position feedback (PPF) control and direct velocity feedback control [3, 4]. Cupiał [5] proposed an optimization problem for finding the control voltage applied to a single piezoelectric patch actuator or several actuators for suppressing vibrations of a beam or rectangular plate. Yiqi and Yiming [6] established an analytical model for the active vibration control of a piezoelectric FGM plate based on a higher-order shear deformation plate and elastic piezoelectric theories. He et al. [7] presented a finite element formulation for the vibration control of a functionally graded plate (FGP) based on the classical lamination plate theory (CLPT). In the previous works of [6, 7], the plates were fully covered with integrated piezoelectric sensors and actuators and they employed piezoelectric patches. Thus, in these works, the optimization procedure was not required for an active vibration control. Orszulik and Shan [8] presented a system identification and vibration control strategy for a flexible manipulator with a collocated piezoelectric sensor/actuator pair. However, they did not perform any optimization on the location of piezoelectric transducers. Julai and Tokhi [9] presented an active vibration control mechanism using GA and particle swarm optimization. However, the optimization procedure for sensor/actuator locations was not performed in their work. Kulkarni et al. [10] presented numerical and experimental studies on active control of a coupled bending-torsion vibration modes of a typical wing box structure of an aircraft using piezoelectric ceramic stack actuators. Zhao and Hu [11] developed a simplified method for analyzing and designing against alleviating the effect of buffeting in vertical tail system. In [10, 11], an optimization procedure was not performed and the formulation would be different from that formulated in the present study.Kumar and Narayanan [12] determined the optimal locations of the sensor/actuator pair on a flexible beam. In this work, they used the linear quadratic regulator (LQR) strategy for active vibration control. Bruant et al. [13] proposed a method based on minimizing the mechanical energy integral of the system for obtaining the optimal actuator locations and maximizing the energy of the state outputs for obtaining the optimal sensor locations on the flexible beam. Hać and Liu [14] derived analytical modeling of a smart plate equipped with piezoelectric sensors and actuators. In their work, two criteria for the optimal configuration of piezoelectric actuators were suggested using modal controllability and the controllability grammian.Bruant et al. [15] developed a simple tool to simulate the active control of laminated plates. In a next step, they optimized the geometry and the number of sensors and actuators. Halim and Moheimani [16] suggested a criterion for finding the optimal placement of collocated piezoelectric actuator/sensor pairs on a thin flexible plate. However, in this work, not only the equations of motion were not derived analytically but also the control algorithm had not been used for suppressing the plate vibration. Yang and Lee [17] developed an integer-real-encoded GA to search for the optimal placement and size of the piezoelectric patches. Rader et al. [18] presented an approach to optimizing the configuration of piezoelectric actuators for vibration control of a flexible aircraft fin. However, not only they derived the equations of motion of smart structure explicitly, but also the residual modes were not considered in their work. Ambrosio et al. [19] presented an H2 norm approach for the actuator and sensor placement. Sajizadeh and Darus [20] determined the optimal locations of sensors based on observability concept. Qian et al. [21] formulated the optimal actuator/sensor location problem in a framework of a zero-one optimization problem, which could be solved using GA. However, in this work, residual modes were neglected. Thus, spillover effects could distort the system performance. Bruant et al. [22] formulated the optimal locations and number of piezoelectric sensors for active vibration control. However, in [20, 22], they did not find the optimal locations of actuators.The spillover effect could be among one of the most challenging issues that might spoil the active vibration control of structure. Indeed, when feedback controllers would be used for active vibration control of systems, there would be no assurance that control algorithm could excite the residual modes. Thus, for enhancing control algorithm operation, a consideration of residual modes should be necessary. However, in [14, 23, 24], residual modes were neglected. Thus, spillover effects could distort the system operation. Bruant et al. [25] and Han and Lee [26] determined the sensor and actuator locations with the consideration of controllability, observability, and spillover reduction. Furthermore, they used GA to find efficient locations of piezoelectric sensors/actuators. However, in their papers, the dynamic characteristics of both rectangular plate and piezoelectric sensors/actuators were not derived explicitly. Bruant and Proslier [27] presented an optimization problem based on optimization criteria of ensuring either good observability or good controllability of the structure, as well as considering residual modes to limit the spillover effects. However, in [27], the dynamic characteristics of both rectangular plate and piezoelectric sensors/actuators were not derived explicitly.In the present paper, an integrated and consistent formulation is derived based on energy principles and variational methods for both the host plate structure and the patch transducers, together with a GA optimization algorithm for optimal locations and orientations of these patches using spatial controllability-/observability-based optimality criteria. In addition, in this study, the effects of residual modes are incorporated in the fitness function of optimization algorithm. Based on the CPT and linear piezoelectric theory, the equations of motion and sensor output equations of the smart plate are derived by using Hamilton’s principle, Rayleigh-Ritz approximation procedure, and the assumed modes method. For increasing the system efficiency, the optimal locations and orientations of piezoelectric transducers are determined based on spatial controllability and observability and considering residual modes to reduce the spillover effect. For active vibration control, a negative velocity feedback control algorithm is used.The remainder of the paper is organized as follows. We derive the basic equation in Section2. In Section 3, we discuss the optimization of the location of sensors and actuators mounted on a rectangular plate. The GA is used to find the optimal location of sensors and actuators in Section 4. In Section 5, designing controller for active vibration of plate and several simulations for showing the influence of optimization and active control algorithm are outlined. Finally, in Section 6, we draw conclusions.


## 2. Basic Equations

### 2.1. Potential and Kinetic Energies and Virtual Work

Consider a flexible plate (Figure1) with N
a piezoelectric actuators and N
se piezoelectric sensors. The following formulation could be used for any plate geometry and any boundary conditions. The total potential energy of the structure and piezoelectric patch is expressed as [28]
(1)
PE
=
∫
V
st
1
2
S
T
c
s
t
S
d
V
st
+
∑
j
=
1
N
a
∫
V
j
a
(
1
2
S
~
T
c
D
S
~
-
S
~
T
h
D
~
j
+
1
2
D
~
j
T
β
S
D
~
j
)
d
V
j
a
+
∑
j
=
1
N
se
∫
V
j
se
(
1
2
S
~
T
c
D
S
~
-
S
~
T
h
D
~
j
+
1
2
D
~
j
T
β
S
D
~
j
)
d
V
j
se
,
where S
~ and D
~
j are strain and electric charge density vector in local coordinates, respectively. c
s
t, c
D, h, and β
S are the matrices of elastic constants of structure, elastic constants of piezoelectric patches under constant electric charge density, the piezoelectric constants, and the inverse of dielectric constant under constant strain; S
~
T and D
~
j
T are the transposes of S
~ and D
~
j. V
st, V
a, and V
se are the volumes of structure, piezoelectric actuators, and piezoelectric sensors, respectively. N
a and N
se are the number of actuator and sensor patches. By using the following relation, the local coordinate of system for each piezoelectric element can be transformed to global coordinates:
(2)
S
~
=
R
S
j
S
,
D
~
j
=
R
D
j
D
j
,
where R
S
j and R
D
j are strain and electric charge density transformation matrices, respectively. The total kinetic energy of the structure and piezoelectric patches are obtained as [28]
(3)
KE
=
1
2
∫
V
st
ρ
st
(
u
˙
1
2
+
u
˙
2
2
+
u
˙
3
2
)
d
V
st
+
1
2
∑
j
=
1
N
a
∫
V
j
a
ρ
j
a
(
u
˙
1
2
+
u
˙
2
2
+
u
˙
3
2
)
d
V
j
a
+
1
2
∑
j
=
1
N
se
∫
V
j
se
ρ
j
se
(
u
˙
1
2
+
u
˙
2
2
+
u
˙
3
2
)
d
V
j
se
=
1
2
∫
V
st
ρ
st
(
u
˙
T
u
˙
)
d
V
st
+
1
2
∑
j
=
1
N
a
∫
V
j
a
ρ
j
a
(
u
˙
T
u
˙
)
d
V
j
a
+
1
2
∑
j
=
1
N
se
∫
V
j
se
ρ
j
se
(
u
˙
T
u
˙
)
d
V
j
se
,
where u
˙
1, u
˙
2, and u
˙
3 are the velocity components in the x, y, and z directions and ρ
st, ρ
a, and ρ
se are the mass densities of structure, piezoelectric actuators, and piezoelectric sensors, respectively. The virtual work of external forces is written as [28]
(4)
δ
W
ext
=
∑
j
=
1
N
a
v
j
a
δ
q
j
a
+
∑
j
=
1
N
se
v
j
se
δ
q
j
se
+
∑
j
=
1
N
f
δ
u
(
x
j
,
y
j
)
T
f
j
,
where v and f are voltage that applied to piezoelectric patches and concentrated force and δ
q and δ
u are variation of electric charge and mechanical displacement.Figure 1
Flexible plate and orientation of piezoelectric patches.


### 2.2. Discretized Displacement Field

In the Rayleigh-Ritz method, the spatial displacement field of midplane for a vibrating rectangular plate is written as(5)
u
1
(
x
,
y
,
t
)
=
N
u
1
(
x
,
y
)
U
1
(
t
)
,
u
2
(
x
,
y
,
t
)
=
N
u
2
(
x
,
y
)
U
2
(
t
)
,
u
3
(
x
,
y
,
t
)
=
N
u
3
(
x
,
y
)
U
3
(
t
)
,
where u
1, u
2, and u
3 are the midplane displacements of the plate along the x, y, and z directions, respectively. U
1, U
2, and U
3 are the generalized coordinates of plate response, and N
u
1, N
u
2, and N
u
3 are the shape function that need to satisfy geometric boundary conditions. Equation (5) can be expressed as a matrix relationship:
(6)
u
=
{
u
1
u
2
u
3
}
=
[
N
u
1
0
0
0
N
u
2
0
0
0
N
u
3
]
{
U
1
U
2
U
3
}
=
N
u
U
.


### 2.3. Discretized Strain-Displacement Relationship

The relationship between strain and displacement, based on the classical plate theory, is expressed as [28]
(7)
{
S
x
x
S
y
y
S
x
y
}
=
{
∂
u
1
∂
x
∂
u
2
∂
y
∂
u
1
∂
y
+
∂
u
2
∂
x
}
+
z
{
-
∂
2
u
3
∂
x
2
-
∂
u
3
∂
x
-
2
∂
2
u
3
∂
x
∂
y
}
.In terms of generalized coordinates, (7) becomes
(8)
{
S
x
x
S
y
y
S
x
y
}
=
[
∂
N
u
1
∂
x
0
0
0
∂
N
u
2
∂
y
0
∂
N
u
1
∂
y
∂
N
u
2
∂
x
0
]
{
U
1
U
2
U
3
}
+
[
0
0
-
z
∂
2
N
u
3
∂
x
2
0
0
-
z
∂
2
N
u
3
∂
y
2
0
0
-
z
∂
2
N
u
3
∂
x
∂
y
]
{
U
1
U
2
U
3
}
=
B
U
m
U
+
B
U
b
U
,
where B
U
m and B
U
b are, respectively, strain-displacement transformation matrices for membrane and flexural actions.


### 2.4. Discretized Electric Charge Density

In this subsection, we attempt to discrete the electric charge density. For achieving this purpose, we assume that the electric charge density is constant within a layer of a patch. Additionally, we assume that electric charge density acts only in the thickness direction of piezoelectric patches and the electric charge on each patch is chosen as a generalized coordinate. With this assumption, a mathematical relationship between electric charge density and the corresponding generalized coordinate for each piezoelectric patchj is written as [28]
(9)
D
j
=
B
q
j
q
p
⟹
B
q
1
=
[
0
0
0
0
0
⋯
0
1
A
p
1
0
0
]
,
B
q
2
=
[
0
0
0
0
0
⋯
0
0
1
A
p
2
0
]
,
B
q
N
p
=
[
0
0
0
0
0
⋯
0
0
0
1
A
p
N
p
]
,
where A is the area of piezoelectric patches. By using (9), the electrical generalized coordinate vector is defined as
(10)
q
p
=
[
q
1
p
q
2
p
q
3
p
⋮
q
N
p
p
]
,
where p
=
a or p
=
se. In this content, a means actuator and se means sensor. Inserting (8) and (9) into the equation of potential energy, that is, (1), yields the following expression that is a function of generalized coordinates [28]:
(11)
PE
=
1
2
U
T
K
s
t
U
+
1
2
U
T
K
D
s
e
U
+
1
2
U
T
K
D
a
U
-
U
T
K
U
q
s
e
q
s
e
-
U
T
K
U
q
a
q
a
+
1
2
q
s
e
T
K
q
q
s
e
q
s
e
+
1
2
q
a
T
K
q
q
a
q
a
,
where K
s
t, K
D
s
e, and K
D
a are, respectively, the stiffness matrices of the structure, piezoelectric sensors, and piezoelectric actuators, K
U
q
s
e and K
U
q
a are coupling matrices of sensors and actuators, and K
q
q
s
e and K
q
q
a are capacitance matrices of sensors and actuators, respectively. The matrices are calculated as
(12)
K
s
t
=
∫
V
st
(
B
U
m
T
c
s
t
B
U
m
+
B
U
b
T
c
s
t
B
U
b
+
B
U
m
T
c
s
t
B
U
b
hhhhh
+
B
U
b
T
c
s
t
B
U
m
)
d
V
st
,
K
D
p
=
∑
j
=
1
N
p
∫
V
j
p
(
B
U
m
T
R
S
j
T
c
D
R
S
j
B
U
m
+
B
U
b
T
R
S
j
T
c
D
R
S
j
B
U
b
hhhh
.
hhh
+
B
U
m
T
R
S
j
T
c
D
R
S
j
B
U
b
+
B
U
b
T
R
S
j
T
c
D
R
S
j
B
U
m
)
hhhhh
×
d
V
j
p
,
K
U
q
p
=
∑
j
=
1
N
p
(
∫
V
j
p
B
U
m
T
R
S
j
T
h
R
D
j
B
q
j
d
V
j
p
h
h
h
h
+
∫
V
j
p
B
U
b
T
R
S
j
T
h
R
D
j
B
q
j
d
V
j
p
)
,
K
q
q
p
=
∑
j
=
1
N
p
∫
V
j
p
B
q
j
T
R
D
j
T
β
s
R
D
j
B
q
j
d
V
j
p
,
where p
=
a or p
=
se. If we insert (6) into (3), kinetic energy can be rewritten as
(13)
KE
=
1
2
U
˙
M
s
t
U
˙
+
1
2
U
˙
M
s
e
U
˙
+
1
2
U
˙
M
a
U
˙
,
where M
s
t, M
s
e, and M
a are mass matrices of the plate, piezoelectric sensors, and piezoelectric actuators. The matrices are calculated as
(14)
M
s
t
=
∫
V
st
ρ
st
N
u
T
N
u
d
V
st
,
M
p
=
∑
j
=
1
N
p
∫
V
j
p
ρ
j
p
N
u
T
N
u
d
V
j
p
,
where p
=
a or p
=
se.By inserting (6) into (4), the external work can be written as
(15)
δ
W
ext
=
δ
q
a
T
B
V
a
V
a
+
δ
q
s
e
T
B
V
s
e
V
s
e
+
δ
U
T
F
c
,
where F
c is
(16)
F
c
=
∑
j
=
1
N
f
N
u
T
(
x
j
,
y
j
)
f
j
.
In addition, B
V
a and B
V
s
e are defined as
(17)
B
v
a
=
I
N
a
,
B
v
s
e
=
I
N
s
e
,
whereI is an identity matrix.


### 2.5. Governing Equations of Motion

Utilizing Hamilton’s principle, the governing equations for the plate vibration equipped with piezoelectric sensor and actuator patches can be derived:(18)
M
U
¨
+
K
U
U
U
-
K
U
q
s
e
q
s
e
-
K
U
q
a
q
a
=
F
,
(19)
-
K
q
U
s
e
U
+
K
q
q
s
e
q
s
e
=
B
v
s
e
v
(
t
)
s
e
,
(20)
-
K
q
U
a
U
+
K
q
q
a
q
a
=
B
v
a
v
(
t
)
a
,
where K
U
q
=
K
q
U
T
,
M
=
M
s
t
+
M
s
e
+
M
a, and K
U
U
=
K
s
t
+
K
s
e
+
K
a. Further, v
(
t
)
a and v
(
t
)
s
e are the external voltages applied to the actuators and sensors. M and K
U
U are, respectively, the total mass and passive stiffness matrices of the system and (19) can be rewritten as
(21)
q
s
e
=
K
q
q
s
e
-
1
B
v
s
e
v
s
e
+
K
q
q
s
e
-
1
K
q
U
s
e
U
.
It is supposed that there is no external applied voltage across the sensor patches; thus v
s
e
=
0. Doing the above operation for (20) results in actuator electric charge:
(22)
q
a
=
K
q
q
a
-
1
B
v
a
v
a
+
K
q
q
a
-
1
K
q
U
a
U
.
Substituting (21) and (22) into (18) yields the following expression for the equations of motion of the plate:
(23)
M
U
¨
+
K
U
=
Υ
a
v
a
+
F
c
,
where
(24)
Υ
a
=
K
U
q
a
K
q
q
a
-
1
B
v
a
is the influence matrix of input voltage applied across actuator patches. The total active and passive matrix of system stiffness is expressed as follows:
(25)
K
=
K
U
U
-
K
U
q
s
e
K
q
q
s
e
-
1
K
q
U
s
e
-
K
U
q
a
K
q
q
a
-
1
K
q
U
a
.
The current through the thickness of each sensor patch is equal to the time derivative of the electric charge accumulated on the surface of each sensor as given in (21):
(26)
i
s
e
(
t
)
=
d
q
s
e
(
t
)
d
t
.
Whenever a piezoelectric sensor is used for sensing a strain rate, the current can be converted into the output open circuit sensor voltage [3]:
(27)
Φ
s
e
(
t
)
=
G
c
i
(
t
)
,
where G
c is the matrix of constant gains of the current amplifier and converts the sensor current vector to an output sensor voltage vector. Using (21) and (26), the output sensor voltage can be expressed as
(28)
Φ
s
e
(
t
)
=
G
c
K
q
q
s
e
-
1
K
q
U
s
e
U
˙
=
C
U
˙
,
where (28) is the sensor equation, which transforms the strain rate to a voltage.


### 2.6. Modal Equations

The eigenvalue problem of (23) can be solved for determining the eigenvalues and eigenvectors. The orthonormality property must be satisfied as
(29)
H
T
M
H
=
I
,
H
T
K
H
=
Λ
n
2
,
where n is the number of the modes, Λ
n
2
=
diag
⁡
(
[
ω
1
2
ω
2
2
ω
3
2
⋯
ω
n
2
]
) is the eigenvalue matrix, and H is the eigenvector matrix. The modal coordinates are introduced as
(30)
U
=
H
R
.
R is generalized displacement vector. By using modal coordinates, modal equations of motion and modal sensor equation can be obtained as
(31)
R
¨
+
Λ
n
2
R
=
Υ
a
′
v
a
+
H
T
F
c
,
Φ
s
e
(
t
)
=
C
′
R
˙
,
where Υ
a
′
=
H
T
Υ
a and C
′
=
C
H. The addition of a structural damping ratio matrix into the modal equations of motion results in the following expressions [29]:
(32)
R
¨
+
2
Z
Λ
n
R
˙
+
Λ
n
2
R
=
Υ
a
′
v
a
+
H
T
F
c
,
where Z is a diagonal matrix of modal damping ratios, Λ
n is a diagonal matrix of natural frequencies, and Λ
n
2 is a diagonal matrix which is equal to the square of natural frequencies. The matrices of Z, Λ
n, and Λ
n
2 are written as
(33)
Z
=
[
ζ
1
0
⋯
0
0
0
ζ
2
0
0
⋮
⋱
⋮
0
0
ζ
n
-
1
0
0
0
⋯
0
ζ
n
]
,
Λ
n
=
[
ω
1
0
⋯
0
0
0
ω
2
0
0
⋮
⋱
⋮
0
0
ω
n
-
1
0
0
0
⋯
0
ω
n
]
.


## 2.1. Potential and Kinetic Energies and Virtual Work

Consider a flexible plate (Figure1) with N
a piezoelectric actuators and N
se piezoelectric sensors. The following formulation could be used for any plate geometry and any boundary conditions. The total potential energy of the structure and piezoelectric patch is expressed as [28]
(1)
PE
=
∫
V
st
1
2
S
T
c
s
t
S
d
V
st
+
∑
j
=
1
N
a
∫
V
j
a
(
1
2
S
~
T
c
D
S
~
-
S
~
T
h
D
~
j
+
1
2
D
~
j
T
β
S
D
~
j
)
d
V
j
a
+
∑
j
=
1
N
se
∫
V
j
se
(
1
2
S
~
T
c
D
S
~
-
S
~
T
h
D
~
j
+
1
2
D
~
j
T
β
S
D
~
j
)
d
V
j
se
,
where S
~ and D
~
j are strain and electric charge density vector in local coordinates, respectively. c
s
t, c
D, h, and β
S are the matrices of elastic constants of structure, elastic constants of piezoelectric patches under constant electric charge density, the piezoelectric constants, and the inverse of dielectric constant under constant strain; S
~
T and D
~
j
T are the transposes of S
~ and D
~
j. V
st, V
a, and V
se are the volumes of structure, piezoelectric actuators, and piezoelectric sensors, respectively. N
a and N
se are the number of actuator and sensor patches. By using the following relation, the local coordinate of system for each piezoelectric element can be transformed to global coordinates:
(2)
S
~
=
R
S
j
S
,
D
~
j
=
R
D
j
D
j
,
where R
S
j and R
D
j are strain and electric charge density transformation matrices, respectively. The total kinetic energy of the structure and piezoelectric patches are obtained as [28]
(3)
KE
=
1
2
∫
V
st
ρ
st
(
u
˙
1
2
+
u
˙
2
2
+
u
˙
3
2
)
d
V
st
+
1
2
∑
j
=
1
N
a
∫
V
j
a
ρ
j
a
(
u
˙
1
2
+
u
˙
2
2
+
u
˙
3
2
)
d
V
j
a
+
1
2
∑
j
=
1
N
se
∫
V
j
se
ρ
j
se
(
u
˙
1
2
+
u
˙
2
2
+
u
˙
3
2
)
d
V
j
se
=
1
2
∫
V
st
ρ
st
(
u
˙
T
u
˙
)
d
V
st
+
1
2
∑
j
=
1
N
a
∫
V
j
a
ρ
j
a
(
u
˙
T
u
˙
)
d
V
j
a
+
1
2
∑
j
=
1
N
se
∫
V
j
se
ρ
j
se
(
u
˙
T
u
˙
)
d
V
j
se
,
where u
˙
1, u
˙
2, and u
˙
3 are the velocity components in the x, y, and z directions and ρ
st, ρ
a, and ρ
se are the mass densities of structure, piezoelectric actuators, and piezoelectric sensors, respectively. The virtual work of external forces is written as [28]
(4)
δ
W
ext
=
∑
j
=
1
N
a
v
j
a
δ
q
j
a
+
∑
j
=
1
N
se
v
j
se
δ
q
j
se
+
∑
j
=
1
N
f
δ
u
(
x
j
,
y
j
)
T
f
j
,
where v and f are voltage that applied to piezoelectric patches and concentrated force and δ
q and δ
u are variation of electric charge and mechanical displacement.Figure 1
Flexible plate and orientation of piezoelectric patches.


## 2.2. Discretized Displacement Field

In the Rayleigh-Ritz method, the spatial displacement field of midplane for a vibrating rectangular plate is written as(5)
u
1
(
x
,
y
,
t
)
=
N
u
1
(
x
,
y
)
U
1
(
t
)
,
u
2
(
x
,
y
,
t
)
=
N
u
2
(
x
,
y
)
U
2
(
t
)
,
u
3
(
x
,
y
,
t
)
=
N
u
3
(
x
,
y
)
U
3
(
t
)
,
where u
1, u
2, and u
3 are the midplane displacements of the plate along the x, y, and z directions, respectively. U
1, U
2, and U
3 are the generalized coordinates of plate response, and N
u
1, N
u
2, and N
u
3 are the shape function that need to satisfy geometric boundary conditions. Equation (5) can be expressed as a matrix relationship:
(6)
u
=
{
u
1
u
2
u
3
}
=
[
N
u
1
0
0
0
N
u
2
0
0
0
N
u
3
]
{
U
1
U
2
U
3
}
=
N
u
U
.


## 2.3. Discretized Strain-Displacement Relationship

The relationship between strain and displacement, based on the classical plate theory, is expressed as [28]
(7)
{
S
x
x
S
y
y
S
x
y
}
=
{
∂
u
1
∂
x
∂
u
2
∂
y
∂
u
1
∂
y
+
∂
u
2
∂
x
}
+
z
{
-
∂
2
u
3
∂
x
2
-
∂
u
3
∂
x
-
2
∂
2
u
3
∂
x
∂
y
}
.In terms of generalized coordinates, (7) becomes
(8)
{
S
x
x
S
y
y
S
x
y
}
=
[
∂
N
u
1
∂
x
0
0
0
∂
N
u
2
∂
y
0
∂
N
u
1
∂
y
∂
N
u
2
∂
x
0
]
{
U
1
U
2
U
3
}
+
[
0
0
-
z
∂
2
N
u
3
∂
x
2
0
0
-
z
∂
2
N
u
3
∂
y
2
0
0
-
z
∂
2
N
u
3
∂
x
∂
y
]
{
U
1
U
2
U
3
}
=
B
U
m
U
+
B
U
b
U
,
where B
U
m and B
U
b are, respectively, strain-displacement transformation matrices for membrane and flexural actions.


## 2.4. Discretized Electric Charge Density

In this subsection, we attempt to discrete the electric charge density. For achieving this purpose, we assume that the electric charge density is constant within a layer of a patch. Additionally, we assume that electric charge density acts only in the thickness direction of piezoelectric patches and the electric charge on each patch is chosen as a generalized coordinate. With this assumption, a mathematical relationship between electric charge density and the corresponding generalized coordinate for each piezoelectric patchj is written as [28]
(9)
D
j
=
B
q
j
q
p
⟹
B
q
1
=
[
0
0
0
0
0
⋯
0
1
A
p
1
0
0
]
,
B
q
2
=
[
0
0
0
0
0
⋯
0
0
1
A
p
2
0
]
,
B
q
N
p
=
[
0
0
0
0
0
⋯
0
0
0
1
A
p
N
p
]
,
where A is the area of piezoelectric patches. By using (9), the electrical generalized coordinate vector is defined as
(10)
q
p
=
[
q
1
p
q
2
p
q
3
p
⋮
q
N
p
p
]
,
where p
=
a or p
=
se. In this content, a means actuator and se means sensor. Inserting (8) and (9) into the equation of potential energy, that is, (1), yields the following expression that is a function of generalized coordinates [28]:
(11)
PE
=
1
2
U
T
K
s
t
U
+
1
2
U
T
K
D
s
e
U
+
1
2
U
T
K
D
a
U
-
U
T
K
U
q
s
e
q
s
e
-
U
T
K
U
q
a
q
a
+
1
2
q
s
e
T
K
q
q
s
e
q
s
e
+
1
2
q
a
T
K
q
q
a
q
a
,
where K
s
t, K
D
s
e, and K
D
a are, respectively, the stiffness matrices of the structure, piezoelectric sensors, and piezoelectric actuators, K
U
q
s
e and K
U
q
a are coupling matrices of sensors and actuators, and K
q
q
s
e and K
q
q
a are capacitance matrices of sensors and actuators, respectively. The matrices are calculated as
(12)
K
s
t
=
∫
V
st
(
B
U
m
T
c
s
t
B
U
m
+
B
U
b
T
c
s
t
B
U
b
+
B
U
m
T
c
s
t
B
U
b
hhhhh
+
B
U
b
T
c
s
t
B
U
m
)
d
V
st
,
K
D
p
=
∑
j
=
1
N
p
∫
V
j
p
(
B
U
m
T
R
S
j
T
c
D
R
S
j
B
U
m
+
B
U
b
T
R
S
j
T
c
D
R
S
j
B
U
b
hhhh
.
hhh
+
B
U
m
T
R
S
j
T
c
D
R
S
j
B
U
b
+
B
U
b
T
R
S
j
T
c
D
R
S
j
B
U
m
)
hhhhh
×
d
V
j
p
,
K
U
q
p
=
∑
j
=
1
N
p
(
∫
V
j
p
B
U
m
T
R
S
j
T
h
R
D
j
B
q
j
d
V
j
p
h
h
h
h
+
∫
V
j
p
B
U
b
T
R
S
j
T
h
R
D
j
B
q
j
d
V
j
p
)
,
K
q
q
p
=
∑
j
=
1
N
p
∫
V
j
p
B
q
j
T
R
D
j
T
β
s
R
D
j
B
q
j
d
V
j
p
,
where p
=
a or p
=
se. If we insert (6) into (3), kinetic energy can be rewritten as
(13)
KE
=
1
2
U
˙
M
s
t
U
˙
+
1
2
U
˙
M
s
e
U
˙
+
1
2
U
˙
M
a
U
˙
,
where M
s
t, M
s
e, and M
a are mass matrices of the plate, piezoelectric sensors, and piezoelectric actuators. The matrices are calculated as
(14)
M
s
t
=
∫
V
st
ρ
st
N
u
T
N
u
d
V
st
,
M
p
=
∑
j
=
1
N
p
∫
V
j
p
ρ
j
p
N
u
T
N
u
d
V
j
p
,
where p
=
a or p
=
se.By inserting (6) into (4), the external work can be written as
(15)
δ
W
ext
=
δ
q
a
T
B
V
a
V
a
+
δ
q
s
e
T
B
V
s
e
V
s
e
+
δ
U
T
F
c
,
where F
c is
(16)
F
c
=
∑
j
=
1
N
f
N
u
T
(
x
j
,
y
j
)
f
j
.
In addition, B
V
a and B
V
s
e are defined as
(17)
B
v
a
=
I
N
a
,
B
v
s
e
=
I
N
s
e
,
whereI is an identity matrix.


## 2.5. Governing Equations of Motion

Utilizing Hamilton’s principle, the governing equations for the plate vibration equipped with piezoelectric sensor and actuator patches can be derived:(18)
M
U
¨
+
K
U
U
U
-
K
U
q
s
e
q
s
e
-
K
U
q
a
q
a
=
F
,
(19)
-
K
q
U
s
e
U
+
K
q
q
s
e
q
s
e
=
B
v
s
e
v
(
t
)
s
e
,
(20)
-
K
q
U
a
U
+
K
q
q
a
q
a
=
B
v
a
v
(
t
)
a
,
where K
U
q
=
K
q
U
T
,
M
=
M
s
t
+
M
s
e
+
M
a, and K
U
U
=
K
s
t
+
K
s
e
+
K
a. Further, v
(
t
)
a and v
(
t
)
s
e are the external voltages applied to the actuators and sensors. M and K
U
U are, respectively, the total mass and passive stiffness matrices of the system and (19) can be rewritten as
(21)
q
s
e
=
K
q
q
s
e
-
1
B
v
s
e
v
s
e
+
K
q
q
s
e
-
1
K
q
U
s
e
U
.
It is supposed that there is no external applied voltage across the sensor patches; thus v
s
e
=
0. Doing the above operation for (20) results in actuator electric charge:
(22)
q
a
=
K
q
q
a
-
1
B
v
a
v
a
+
K
q
q
a
-
1
K
q
U
a
U
.
Substituting (21) and (22) into (18) yields the following expression for the equations of motion of the plate:
(23)
M
U
¨
+
K
U
=
Υ
a
v
a
+
F
c
,
where
(24)
Υ
a
=
K
U
q
a
K
q
q
a
-
1
B
v
a
is the influence matrix of input voltage applied across actuator patches. The total active and passive matrix of system stiffness is expressed as follows:
(25)
K
=
K
U
U
-
K
U
q
s
e
K
q
q
s
e
-
1
K
q
U
s
e
-
K
U
q
a
K
q
q
a
-
1
K
q
U
a
.
The current through the thickness of each sensor patch is equal to the time derivative of the electric charge accumulated on the surface of each sensor as given in (21):
(26)
i
s
e
(
t
)
=
d
q
s
e
(
t
)
d
t
.
Whenever a piezoelectric sensor is used for sensing a strain rate, the current can be converted into the output open circuit sensor voltage [3]:
(27)
Φ
s
e
(
t
)
=
G
c
i
(
t
)
,
where G
c is the matrix of constant gains of the current amplifier and converts the sensor current vector to an output sensor voltage vector. Using (21) and (26), the output sensor voltage can be expressed as
(28)
Φ
s
e
(
t
)
=
G
c
K
q
q
s
e
-
1
K
q
U
s
e
U
˙
=
C
U
˙
,
where (28) is the sensor equation, which transforms the strain rate to a voltage.


## 2.6. Modal Equations

The eigenvalue problem of (23) can be solved for determining the eigenvalues and eigenvectors. The orthonormality property must be satisfied as
(29)
H
T
M
H
=
I
,
H
T
K
H
=
Λ
n
2
,
where n is the number of the modes, Λ
n
2
=
diag
⁡
(
[
ω
1
2
ω
2
2
ω
3
2
⋯
ω
n
2
]
) is the eigenvalue matrix, and H is the eigenvector matrix. The modal coordinates are introduced as
(30)
U
=
H
R
.
R is generalized displacement vector. By using modal coordinates, modal equations of motion and modal sensor equation can be obtained as
(31)
R
¨
+
Λ
n
2
R
=
Υ
a
′
v
a
+
H
T
F
c
,
Φ
s
e
(
t
)
=
C
′
R
˙
,
where Υ
a
′
=
H
T
Υ
a and C
′
=
C
H. The addition of a structural damping ratio matrix into the modal equations of motion results in the following expressions [29]:
(32)
R
¨
+
2
Z
Λ
n
R
˙
+
Λ
n
2
R
=
Υ
a
′
v
a
+
H
T
F
c
,
where Z is a diagonal matrix of modal damping ratios, Λ
n is a diagonal matrix of natural frequencies, and Λ
n
2 is a diagonal matrix which is equal to the square of natural frequencies. The matrices of Z, Λ
n, and Λ
n
2 are written as
(33)
Z
=
[
ζ
1
0
⋯
0
0
0
ζ
2
0
0
⋮
⋱
⋮
0
0
ζ
n
-
1
0
0
0
⋯
0
ζ
n
]
,
Λ
n
=
[
ω
1
0
⋯
0
0
0
ω
2
0
0
⋮
⋱
⋮
0
0
ω
n
-
1
0
0
0
⋯
0
ω
n
]
.


## 3. Optimal Locations of the Piezoelectric Sensors and Actuators

To determine the optimal piezoelectric sensor and actuator locations for a simply supported plate, a modified optimization criterion based on the concept of spatialℋ
2 norm is used [30]. In this method, two criteria for determining the optimal placement of piezoelectric actuators and sensors are proposed, using spatial controllability/observability and considering the effect of residual modes to reduce the spillover effect for the simply supported plate. Here, we consider a state of no in-plane extension and assume that the plate is only in a condition of pure bending. With this assumption, the spatially discredited displacement field of the midplane is written as
(34)
u
1
=
0
,
u
2
=
0
,
u
3
=
N
u
3
U
3
.


### 3.1. Optimal Placement of Actuators

We assume the voltages applied to the actuator patches are as follows:(35)
v
a
=
[
v
1
a
v
2
a
⋯
v
N
a
a
]
T
.
By the assumption that the initial conditions and external mechanical forces are zero, the transfer function of the plate, resulting from Laplace transforming of (32), can be written as
(36)
G
(
s
,
x
,
y
)
=
∑
i
=
1
n
G
i
(
s
,
x
,
y
)
=
∑
i
=
1
n
N
u
3
i
Υ
a
i
s
2
+
2
ζ
i
ω
i
s
+
ω
i
2
.
Equation (36) is the transfer function from the voltages applied to the actuator patches, v
(
t
)
a, to the plate pure bending, u
3. Υ
a
i is defined as (37)
Υ
a
i
=
[
(
K
U
q
a
(
1
)
)
i
(
K
q
q
a
(
1
)
)
-
1
(
K
U
q
a
(
2
)
)
i
(
K
q
q
a
(
2
)
)
-
1
⋯
(
K
U
q
a
(
N
a
)
)
i
(
K
q
q
a
(
N
a
)
)
-
1
]
,and N
u
3
i is theith component of the vector N
u
3. Now, we are in a position to define the spatial ℋ
2 norm of a transfer function as [28, 31, 32]
(38)
∥
G
∥
2
2
=
∫
-
∞
+
∞
∫
0
l
x
s
∫
0
l
y
s
trace
{
G
(
j
ω
,
x
,
y
)
*
G
(
j
ω
,
x
,
y
)
}
h
h
h
h
h
h
h
h
h
h
h
h
×
d
y
d
x
d
ω
=
∑
i
=
1
n
∥
G
~
i
∥
2
2
,
where trace is defined to be the sum of the elements on the main diagonal of a matrix and
(39)
G
~
i
=
Υ
a
i
s
2
+
2
ζ
i
ω
i
s
+
ω
i
2
,
i
=
1
,
…
,
n
.
We find the optimal location of thejth piezoelectric actuator patch by defining the spatial controllability, function ψ
i
j, as
(40)
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
=
∥
G
~
i
j
∥
2
=
|
(
K
U
q
a
(
j
)
)
i
(
K
q
q
a
(
j
)
)
-
1
|
∥
Υ
a
i
s
2
+
2
ζ
i
ω
i
s
+
ω
i
2
∥
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
=
1
,
…
,
n
,
where (
x
a
(
j
)
,
y
a
(
j
)
) and θ
a
(
j
) are the corner coordinates of thejth actuator patch and its orientation, respectively. The actuator orientation may change in the interval, (
0
≤
θ
<
π
). In this study, the variables of the actuator locations that should be optimized are {
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
},  j
=
1,2
,
…
,
N
a. The patches orientation is shown in Figure 1.For enhancing the performance of feedback control, we should reduce the spillover effects. For achieving this purpose, it is necessary that, with due consideration of the residual modes, we minimize the authority of the actuators over the high-frequency modes. Furthermore, we can enhance the performance of control with considering only the firstN
C modes and place the actuators at the locations that they provide a high-authority over the first N
C modes. The actuators optimal locations can be found by maximizing the following criterion:
(41)
∑
j
=
1
N
a
∑
i
=
1
N
C
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
-
λ
∑
j
=
1
N
a
∑
i
=
N
C
N
C
+
N
R
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
,
where λ is a weighting constant. N
C and N
R are number of controlled and residual modes. The terms, ∑
i
=
1
N
C
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2 and ∑
i
=
N
C
N
C
+
N
R
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2, are, respectively, the spatial controllability of the first N
C modes and the remaining residual modes. Regarding that the spatial controllability is the same as spatial ℋ
2 norm, for obtaining it, a limited number of modes are considered. The optimization criterion for locating actuators can be normalized as
(42)
J
a
=
∑
j
=
1
N
a
∑
i
=
1
N
C
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
max
⁡
∑
j
=
1
N
a
∑
i
=
1
N
C
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
-
λ
∑
j
=
1
N
a
∑
i
=
N
C
N
C
+
N
R
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
max
⁡
∑
j
=
1
N
a
∑
i
=
N
C
N
C
+
N
R
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
.
In (42), since the components of the numerator do not have the same numerical range, normalizing all of them to the same value, that is, the maximum value, is performed. If this normalization procedure would not have been taken into account, the resulting locations and orientations could not have been optimized correctly.


### 3.2. Optimal Placement of Sensors

For optimizing piezoelectric sensor locations, we rewrite (32) as
(43)
Φ
se
(
t
)
=
C
′
R
˙
=
C
U
˙
=
∑
i
=
1
n
C
i
U
˙
i
,
where C
i is defined as (44)
C
i
=
[
G
c
1
(
K
q
q
se
(
1
)
)
-
1
(
K
U
q
se
(
1
)
)
i
G
c
2
(
K
q
q
se
(
2
)
)
-
1
(
K
U
q
se
(
2
)
)
i
⋯
G
c
N
se
(
K
q
q
se
(
N
se
)
)
-
1
(
K
U
q
se
(
N
se
)
)
i
]
T
.We find the optimal location of thejth piezoelectric sensor by defining the spatial observability, function Θ
i
j, as
(45)
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
=
|
C
i
j
|
=
|
G
c
j
(
K
q
q
se
(
j
)
)
-
1
(
K
U
q
se
(
j
)
)
i
|
,
where (
x
se
(
j
)
,
y
se
(
j
)
), and θ
se
(
j
) are the corner coordinates of the jth sensor and its orientation, respectively. The orientation of sensors changes in the interval (
0
≤
θ
<
π
). In this work, the variables of sensor locations that should be optimized are as {
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
}, j
=
1,2
,
…
,
N
se.The optimal placement of sensors can be found by maximizing the following criterion:(46)
∑
j
=
1
N
se
∑
i
=
1
N
C
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
-
λ
~
∑
j
=
1
N
se
∑
i
=
N
C
N
C
+
N
R
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
which can be normalized as
(47)
J
se
=
∑
j
=
1
N
se
∑
i
=
1
N
C
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
max
⁡
∑
j
=
1
N
se
∑
i
=
1
N
C
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
-
λ
~
∑
j
=
1
N
se
∑
i
=
N
C
N
C
+
N
R
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
max
⁡
∑
j
=
1
N
se
∑
i
=
N
C
N
C
+
N
R
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
.
By normalizing each Rayleigh-Ritz mode with respect to corresponding maximal energy that could be achieved from its corresponding mode, the criterion would search for the optimal configuration. In this criterion, the residual modes are considered and their influence on the optimization criterion is controlled by weighting constant, λ
~. By comparing (42) and (47), it is understood that, for optimizing both sensor and actuator locations, the same procedure should be implemented.


## 3.1. Optimal Placement of Actuators

We assume the voltages applied to the actuator patches are as follows:(35)
v
a
=
[
v
1
a
v
2
a
⋯
v
N
a
a
]
T
.
By the assumption that the initial conditions and external mechanical forces are zero, the transfer function of the plate, resulting from Laplace transforming of (32), can be written as
(36)
G
(
s
,
x
,
y
)
=
∑
i
=
1
n
G
i
(
s
,
x
,
y
)
=
∑
i
=
1
n
N
u
3
i
Υ
a
i
s
2
+
2
ζ
i
ω
i
s
+
ω
i
2
.
Equation (36) is the transfer function from the voltages applied to the actuator patches, v
(
t
)
a, to the plate pure bending, u
3. Υ
a
i is defined as (37)
Υ
a
i
=
[
(
K
U
q
a
(
1
)
)
i
(
K
q
q
a
(
1
)
)
-
1
(
K
U
q
a
(
2
)
)
i
(
K
q
q
a
(
2
)
)
-
1
⋯
(
K
U
q
a
(
N
a
)
)
i
(
K
q
q
a
(
N
a
)
)
-
1
]
,and N
u
3
i is theith component of the vector N
u
3. Now, we are in a position to define the spatial ℋ
2 norm of a transfer function as [28, 31, 32]
(38)
∥
G
∥
2
2
=
∫
-
∞
+
∞
∫
0
l
x
s
∫
0
l
y
s
trace
{
G
(
j
ω
,
x
,
y
)
*
G
(
j
ω
,
x
,
y
)
}
h
h
h
h
h
h
h
h
h
h
h
h
×
d
y
d
x
d
ω
=
∑
i
=
1
n
∥
G
~
i
∥
2
2
,
where trace is defined to be the sum of the elements on the main diagonal of a matrix and
(39)
G
~
i
=
Υ
a
i
s
2
+
2
ζ
i
ω
i
s
+
ω
i
2
,
i
=
1
,
…
,
n
.
We find the optimal location of thejth piezoelectric actuator patch by defining the spatial controllability, function ψ
i
j, as
(40)
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
=
∥
G
~
i
j
∥
2
=
|
(
K
U
q
a
(
j
)
)
i
(
K
q
q
a
(
j
)
)
-
1
|
∥
Υ
a
i
s
2
+
2
ζ
i
ω
i
s
+
ω
i
2
∥
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
=
1
,
…
,
n
,
where (
x
a
(
j
)
,
y
a
(
j
)
) and θ
a
(
j
) are the corner coordinates of thejth actuator patch and its orientation, respectively. The actuator orientation may change in the interval, (
0
≤
θ
<
π
). In this study, the variables of the actuator locations that should be optimized are {
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
},  j
=
1,2
,
…
,
N
a. The patches orientation is shown in Figure 1.For enhancing the performance of feedback control, we should reduce the spillover effects. For achieving this purpose, it is necessary that, with due consideration of the residual modes, we minimize the authority of the actuators over the high-frequency modes. Furthermore, we can enhance the performance of control with considering only the firstN
C modes and place the actuators at the locations that they provide a high-authority over the first N
C modes. The actuators optimal locations can be found by maximizing the following criterion:
(41)
∑
j
=
1
N
a
∑
i
=
1
N
C
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
-
λ
∑
j
=
1
N
a
∑
i
=
N
C
N
C
+
N
R
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
,
where λ is a weighting constant. N
C and N
R are number of controlled and residual modes. The terms, ∑
i
=
1
N
C
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2 and ∑
i
=
N
C
N
C
+
N
R
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2, are, respectively, the spatial controllability of the first N
C modes and the remaining residual modes. Regarding that the spatial controllability is the same as spatial ℋ
2 norm, for obtaining it, a limited number of modes are considered. The optimization criterion for locating actuators can be normalized as
(42)
J
a
=
∑
j
=
1
N
a
∑
i
=
1
N
C
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
max
⁡
∑
j
=
1
N
a
∑
i
=
1
N
C
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
-
λ
∑
j
=
1
N
a
∑
i
=
N
C
N
C
+
N
R
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
max
⁡
∑
j
=
1
N
a
∑
i
=
N
C
N
C
+
N
R
ψ
i
j
(
x
a
(
j
)
,
y
a
(
j
)
,
θ
a
(
j
)
)
2
.
In (42), since the components of the numerator do not have the same numerical range, normalizing all of them to the same value, that is, the maximum value, is performed. If this normalization procedure would not have been taken into account, the resulting locations and orientations could not have been optimized correctly.


## 3.2. Optimal Placement of Sensors

For optimizing piezoelectric sensor locations, we rewrite (32) as
(43)
Φ
se
(
t
)
=
C
′
R
˙
=
C
U
˙
=
∑
i
=
1
n
C
i
U
˙
i
,
where C
i is defined as (44)
C
i
=
[
G
c
1
(
K
q
q
se
(
1
)
)
-
1
(
K
U
q
se
(
1
)
)
i
G
c
2
(
K
q
q
se
(
2
)
)
-
1
(
K
U
q
se
(
2
)
)
i
⋯
G
c
N
se
(
K
q
q
se
(
N
se
)
)
-
1
(
K
U
q
se
(
N
se
)
)
i
]
T
.We find the optimal location of thejth piezoelectric sensor by defining the spatial observability, function Θ
i
j, as
(45)
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
=
|
C
i
j
|
=
|
G
c
j
(
K
q
q
se
(
j
)
)
-
1
(
K
U
q
se
(
j
)
)
i
|
,
where (
x
se
(
j
)
,
y
se
(
j
)
), and θ
se
(
j
) are the corner coordinates of the jth sensor and its orientation, respectively. The orientation of sensors changes in the interval (
0
≤
θ
<
π
). In this work, the variables of sensor locations that should be optimized are as {
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
}, j
=
1,2
,
…
,
N
se.The optimal placement of sensors can be found by maximizing the following criterion:(46)
∑
j
=
1
N
se
∑
i
=
1
N
C
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
-
λ
~
∑
j
=
1
N
se
∑
i
=
N
C
N
C
+
N
R
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
which can be normalized as
(47)
J
se
=
∑
j
=
1
N
se
∑
i
=
1
N
C
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
max
⁡
∑
j
=
1
N
se
∑
i
=
1
N
C
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
-
λ
~
∑
j
=
1
N
se
∑
i
=
N
C
N
C
+
N
R
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
max
⁡
∑
j
=
1
N
se
∑
i
=
N
C
N
C
+
N
R
Θ
i
j
(
x
se
(
j
)
,
y
se
(
j
)
,
θ
se
(
j
)
)
2
.
By normalizing each Rayleigh-Ritz mode with respect to corresponding maximal energy that could be achieved from its corresponding mode, the criterion would search for the optimal configuration. In this criterion, the residual modes are considered and their influence on the optimization criterion is controlled by weighting constant, λ
~. By comparing (42) and (47), it is understood that, for optimizing both sensor and actuator locations, the same procedure should be implemented.


## 4. Application of GA for Optimal Location of Piezoelectric Devices

In this study, the GA is utilized for optimizing the piezoelectric device locations on the flexible plate. GA is a means by which the machine can simulate the mechanism of natural selection. This operation is done by searching in the design space for finding the best solution. The GA starts a search from a series of points, and, for performing the search procedure, it does not require the Jacobian of functions [33]. Many researchers for optimizing their work have used the GA [34–37]. The terminology used for GA is brought in Table 1.Table 1
Terminology for GA.
Terminology
Description
Fitness function
The function that should be optimized (J in (42) and (47)).
Individual
Any point that is utilized in the fitness function is an individual. The number of variables for finding piezoelectric device positions and orientations is either3
×
N
se or 3
×
N
a that is equal to the number of individuals.
Population
An array of individuals forms the population. If the size of population isN
i and the number of variables or individuals is 3
×
N
se or 3
×
N
a, then the population is shown by N
i by 3
×
N
se or N
i by 3
×
N
a matrices.
Beginning
Creating a casual population is the first step of GA.
Evaluation
The value ofJ in (42) and (47) to be evaluated for each population.
Selection
Select two parents having the largest value of the fitness function.
Crossover
Incorporate two parents that produce new children.
Mutation
In order that this operator produces the children, it applies a random change in parents. New children are placed in new populationUsually with the progress in the next generations, the algorithm converges to an asymptote point and the amount of fitness function improves. When the convergence is reached or the stopping criteria are satisfied, the search process stops. For optimization, a GA with the following configuration is considered in the work: population size: 200, crossover rate: 0.8, and number of generations: 50.Numerical simulations of the optimization process are analyzed in this section. We consider a simply supported rectangular plate for which the piezoelectric patches are attached to both upper and lower surfaces. The mechanical properties of aluminum that is used in this paper areρ
=
2770 kg/m3, v
=
0.3, E
=
70 GPa, and ζ
i = 0.0002. We suppose that the piezoelectric sensors and actuators are perfectly bonded to the surfaces and they are made from the same piezoelectric materials and symmetrically attached to the plate. Tables 2 and 3 give the geometrical and mechanical properties of materials that are used in this study.Table 2
Geometrical properties of the plate and the piezoelectric sensors and actuators.
Plate
Piezoelectric patch
x-length, l
x
s (m) and l
x (m)
1
0.08
y-length, l
y
s (m) and l
y (m)
0.5
0.04
Thickness,h
s
t and h
p (m)
0.002
0.0001Table 3
Material properties of piezoelectric transducers made of PZT-5H from [28].
Mechanical properties
c
11
D
=
c
22
D
=
131.6 GPa
c
12
D
=
84.2 GPa
c
66
D
=
3 GPa
ρ
=
7800 kg/m3
Electrical property
β
33
s
=
1.48
×
1
0
8 m/F
Coupling coefficients
h
13
=
h
23
=
-
2.72
×
1
0
9 N/CBy using the Rayleigh–Ritz method and expanding the transverse displacement of the plate in terms of a time dependent modal shape series, satisfying the geometric boundary conditions, we have:(48)
u
3
(
x
,
y
,
t
)
=
∑
κ
=
1
k
∑
ι
=
1
l
N
u
3
κ
ι
(
x
,
y
)
U
κ
ι
(
t
)
,
where k and l are the total number of mode shapes in the longitudinal x and lateral y directions, respectively.  κ and ι are the number of half-waves in the longitudinal x and lateral y directions; this series can be expressed as a matrix expansion:
(49)
u
3
=
[
N
u
3
11
N
u
3
21
N
u
3
31
⋯
N
u
3
k
l
]
[
U
11
U
21
U
31
⋯
U
k
l
]
T
.
We define
(50)
N
u
3
=
[
N
u
3
11
N
u
3
21
N
u
3
31
⋯
N
u
3
k
l
]
.
The boundary conditions of plate are simply supported:
(51)
u
1
=
u
2
=
u
3
=
0
at
x
=
0
,
x
=
l
x
s
,
y
=
0
,
y
=
l
y
s
,
where l
x
s and l
y
s are the dimensions of rectangular plate.For a simply supported plate, the modal shape functions that are used in this study are expanded in terms of double harmonic functions:(52)
N
u
3
κ
ι
=
sin
(
κ
π
x
l
x
s
)
sin
(
ι
π
y
l
y
s
)
,
where l
x
s and l
y
s are the length and width of rectangular plate. For the purpose of vibration control, only the first five modes are considered that their natural frequencies are presented in Table 4. We have taken into account the influences of piezoelectric patches into our general formulation. However, we have neglected these effects in our solution for a simply supported rectangular plate, since the dimensions of piezoelectric patches are small in comparison with the host plate. We neglected their influences in calculating the natural frequencies and eigenmodes. Thus, in (18), M
=
M
s
t and K
U
U
=
K
s
t. There are several examples considered here for showing the influence of GA that is used for locating the piezoelectric devices attached to a thin plate. At first, we discuss the values of λ in the optimization procedure. The different values of λ that are substituted in (42) for the case of a single actuator and sensor are listed in Tables 5–7. In Table 5, the first mode is taken into account that would have the highest controllability, and else the next four modes are considered as the residual modes. As can be seen, when the values of λ increase, the values of (40) decrease and the controllability decreases. It means that, for large values of λ, GA could not find the optimal position and orientation of the actuator. In Table 6, the second mode is taken into account that would have the highest controllability, and else the other modes are considered as the residual modes. As can be understood, similar to previous case, when the values of λ increase, the values which appeared in (40) decrease and the controllability decreases, meaning that, for large values of λ, GA cannot find the optimal position and orientation of the actuator. In Table 7, the first two modes are unresidual modes, and else the other three modes are considered as the residual modes. For this case, several optimal positions are obtained that some of them are the nearly close for the different values of λ. Thus, the selection of λ from disjoint ranges, as presented in Table 7, does not have any observable influence on finding the optimal positions and orientations. Thus, we would choose the value of λ equal to two.Table 4
Natural frequencies of the plate (before active control).
Mode number
1
2
3
4
5
Longitudinal and circumferential modes (κ
,
ι)
(1, 1)
(2, 1)
(3, 1)
(1, 2)
(2, 2)
Natural frequencies [Hz]
23.88
38.22
62.11
81.23
95.57Table 5
The effect of values of the weight coefficients on the controllability for a single actuator for the case that only the first mode is controlled.
Actuator longitudinal coordinatex
a
Actuator lateral coordinatey
a
Actuator orientationθ
a
Weight coefficientλ
Values of controllability ofith mode
1
2
3
4
5
0.487 m
0.207 m
0.818 rad
0.500
0.022
0.000
0.057
0.000
0.000
0.538 m
0.226 m
2.114 rad
0.100
0.022
0.000
0.056
0.000
0.000
0.494 m
0.455 m
1.188 rad
1.000
0.000
0.000
0.000
0.000
0.000
0.469 m
0.457 m
1.350 rad
2.000
0.000
0.000
0.000
0.000
0.000
0.501 m
0.455 m
1.121 rad
3.000
0.000
0.000
0.000
0.000
0.000Table 6
The effect of values of the weight coefficients on the controllability for a single actuator for the case that only the second mode is controlled.
Actuator longitudinal coordinatex
a
Actuator lateral coordinatey
a
Actuator orientationθ
a
Weight coefficientλ
Values of controllability ofith mode
1
2
3
4
5
0.294 m
0.217 m
1.856 rad
0.100
0.012
0.035
0.021
0.000
0.000
0.677 m
0.206 m
1.329 rad
0.500
0.016
0.026
0.000
0.000
0.000
0.489 m
0.455 m
1.104 rad
1.000
0.000
0.000
0.000
0.000
0.000
0.878 m
0.459 m
0.705 rad
2.000
0.000
0.000
0.000
0.000
0.000
0.503 m
0.455 m
1.167 rad
3.000
0.000
0.000
0.000
0.000
0.000Table 7
The effect of values of the weight coefficients on the controllability for a single actuator for the case that the first two modes are controlled.
Actuator longitudinal coordinatex
a
Actuator lateral coordinatey
a
Actuator orientationθ
a
Weight coefficientλ
Values of controllability ofith mode
1
2
3
4
5
0.363 m
0.250 m
2.672 rad
0.100
0.016
0.029
0.001
0.000
0.000
0.670 m
0.205 m
1.178 rad
0.500
0.016
0.026
0.000
0.000
0.000
0.314 m
0.210 m
0.651 rad
2.000
0.016
0.026
0.000
0.000
0.000
0.695 m
0.215 m
1.789 rad
3.000
0.017
0.026
0.000
0.000
0.000
0.702 m
0.223 m
2.021 rad
5.000
0.017
0.026
0.000
0.000
0.000In the first two simulations, the first two lowest modes would seem to have the highest controllability and observability and else the remaining three modes are considered as the residual modes. Because the fitness functions for finding sensor and actuator (transducer) locations and orientations are the same, herein, we only discuss how to find the actuator locations and orientations. In Figure2(a), the best and mean values of the fitness function for finding the optimal location of one transducer attached to the plate are shown. As can be seen, the best value for the objective function is equal to 1.68. The optimal corner point coordinates for a single-transducer are (0.657 m, 0.206 m), while the transducer orientation in this point is 0.9 rad. The best and mean fitness function values for finding the optimal locations of two transducers are indicated in Figure 2(b). In the first generations, the best value of fitness function is improved rapidly that is because of being too far away from the optimal point. In the other simulations of this section, the first three lower modes are controlled and the remaining two modes are considered as residual modes. The evolution of the best and mean values of the fineness function is presented in Figure 3. The optimal positions of these simulations are listed in Tables 8 and 9.Table 8
GA-optimal and random configurations of piezoelectric transducers attached to the rectangular plate. The first two modes are the dominant modes, while the next three modes are the residual modes.
Location type
Optimal locations from GA
Random locations (no optimization)
Actuator longitudinal coordinatex
a
Actuator lateral coordinatey
a
Actuator orientationθ
a
Actuator longitudinal coordinatex
a
Actuator lateral coordinatey
a
Actuator orientationθ
a
Transducer data
Single transducer
0.657 m
0.206 m
0.9 rad
0.757 m
0.386 m
0.905 rad
Double transducers
0.673 m0.378 m
0.205 m0.251 m
1.25 rad2.71 rad
0.757 m0.278 m
0.386 m0.351 m
0.905 rad2.706 radTable 9
GA-optimal and random configurations of piezoelectric transducers attached to the rectangular plate. The first three modes are the dominant modes, while the next two modes are the residual modes.
Location type
Optimal locations from GA
Random locations (no optimization)
Actuator longitudinal coordinatex
a
Actuator lateral coordinatey
a
Actuator orientationθ
a
Actuator longitudinal coordinatex
a
Actuator lateral coordinatey
a
Actuator orientationθ
a
Transducer data
Single transducer
0.481 m
0.227 m
2.14 rad
0.531 m
0.257 m
2.14 rad
Double transducers
0.437 m0.475 m
0.215 m0.152 m
1.8 rad1.49 rad
0.337 m0.275 m
0.115 m0.352 m
1.796 rad1.494 radEvolution of the mean and best values of fitness function for finding actuator-sensor locations when the first two modes are controlled, while the next three modes are the residual modes: (a) one actuator-sensor and (b) a pair of actuator-sensors.
(a)
(b)Evolution of the mean and best values of fitness function for finding actuator-sensor locations when the first three modes are controlled, while the next two modes are the residual modes: (a) one actuator-sensor and (b) a pair of actuator-sensors.
(a)
(b)


## 5. Results and Discussion

### 5.1. Controller Design

For active vibration control, a negative velocity feedback control algorithm is used. This feedback strategy increases the depreciation rate; thus it is an effective way for reducing the oscillatory amplitude. A simple algorithm of this type is used in such a way that the actuator voltage can be obtained with an amplifier gain and a change of polarity on the sensor voltage as follows:(53)
v
(
t
)
a
=
-
G
C
o
Φ
s
e
(
t
)
,
where G
C
o is a matrix of gains of the amplifier that is used in the feedback control.In next subsection, several simulations are presented for illustrating the influence of optimization procedure and active control algorithm.


### 5.2. Simulation 1

In the first simulation, the first two lower modes should be controlled and the remaining three modes are considered as residual modes. The transducers should be located at random and optimal positions that are listed in Table8. For exciting a larger number of system mode shapes, we have applied the step function concentrated load at a location off-center of plate. The frequency responses of the plate, when excited by a concentrated force and located at x
=
0.43
m and y
=
0.21
m for different number of sensors and actuators, are plotted in Figure 4. As can be seen, the optimization procedure is more effective for the amplitude vibration reduction as contributed to arbitrary location of actuators/sensors. By locating the actuators and sensors in the optimal locations, more damping is developed in the plate. When the sensors are located at random locations, we could not observe the first two modes in the response, and when the actuators are positioned at random locations the maximum mechanical energy cannot be transmitted to the first two modes. However if the sensors are located at the optimal locations, the contributed modes could be observed. Furthermore, if the actuators are located at their optimal positions, the maximum mechanical energy could be transmitted to the contributed modes, causing the amplitude of vibration could be reduced more. The values of spatial controllability (40) are shown in Tables 10 and 11. The transducers are located in position of Table 6. The first two modes are the controlled modes and the modes 3–5 are the residual modes. By comparing the values of these two tables, it can be seen that, for the optimal locations, the values of the system controllability for the first two modes are greater than the values of an arbitrarily selected location. Thus, the optimization procedure would seem to be effective. In Figure 5, two actuators are used for active vibration control of the simply supported rectangular plate under consideration. In this figure, we would compare the optimal configurations in Table 8 and use the optimal locations of one actuator activated for the first mode and the other actuator for the second mode. As can be seen, the results based on mathematical procedure used based on GA and represented in Table 8 are more effective than the individual activations of each actuator for each mode.Table 10
The effect of values of controllability for optimal location, whenever the first two modes are controlled.
Values of controllability ofith mode
Number of actuators
1
2
3
4
5
One actuator
0.033
0.053
0.000
0.000
0.000
Two actuators
0.016
0.026
0.000
0.000
0.000Table 11
The effect of values of controllability for random location, whenever the first two modes are controlled.
Values of controllability ofith mode
Number of actuators
1
2
3
4
5
One actuator
0.001
0.004
0.010
0.008
0.032
Two actuators
0.001
0.007
0.018
0.010
0.040Frequency response of the plate controlled by the piezoelectric actuator-sensor patches when the first two modes are controlled, while the next three modes are the residual modes: (a) one actuator-sensor and (b) a pair of actuator-sensors.
(a)
(b)Figure 5
Comparison of the optimal configurations in Table8 and use of the optimal locations of one actuator activated for the first mode and the other actuator for the second mode.


### 5.3. Simulation 2

In the second simulation, the first three lower modes must be controlled and the remaining two modes are residual modes. The transducers should be located at random and optimal positions that are listed in Table9. The frequency responses of the plate, excited by a concentrated force and located at x
=
0.43
m and y
=
0.21
m, for single and double transducer configurations, are presented in Figure 6. As can be seen, the optimization criteria are effective in having more damping effect. In Figure 7, the influences of the number of sensors and actuators are compared in the control of vibration. In this simulation, the sensors and actuators are located in the optimal positions and the first two lower modes must be controlled. It can be concluded that when a pair of actuators and sensors are used the damping effect increases and the amplitude vibration of plate reduces more.Frequency response of the plate controlled by the piezoelectric actuator-sensor. The first three modes are the controlled modes, while the next two modes are the residual modes: (a) one actuator-sensor and (b) two actuator-sensors.
(a)
(b)Figure 7
Comparison of frequency response of the plate controlled by different number of piezoelectric actuator-sensors.


## 5.1. Controller Design

For active vibration control, a negative velocity feedback control algorithm is used. This feedback strategy increases the depreciation rate; thus it is an effective way for reducing the oscillatory amplitude. A simple algorithm of this type is used in such a way that the actuator voltage can be obtained with an amplifier gain and a change of polarity on the sensor voltage as follows:(53)
v
(
t
)
a
=
-
G
C
o
Φ
s
e
(
t
)
,
where G
C
o is a matrix of gains of the amplifier that is used in the feedback control.In next subsection, several simulations are presented for illustrating the influence of optimization procedure and active control algorithm.


## 5.2. Simulation 1

In the first simulation, the first two lower modes should be controlled and the remaining three modes are considered as residual modes. The transducers should be located at random and optimal positions that are listed in Table8. For exciting a larger number of system mode shapes, we have applied the step function concentrated load at a location off-center of plate. The frequency responses of the plate, when excited by a concentrated force and located at x
=
0.43
m and y
=
0.21
m for different number of sensors and actuators, are plotted in Figure 4. As can be seen, the optimization procedure is more effective for the amplitude vibration reduction as contributed to arbitrary location of actuators/sensors. By locating the actuators and sensors in the optimal locations, more damping is developed in the plate. When the sensors are located at random locations, we could not observe the first two modes in the response, and when the actuators are positioned at random locations the maximum mechanical energy cannot be transmitted to the first two modes. However if the sensors are located at the optimal locations, the contributed modes could be observed. Furthermore, if the actuators are located at their optimal positions, the maximum mechanical energy could be transmitted to the contributed modes, causing the amplitude of vibration could be reduced more. The values of spatial controllability (40) are shown in Tables 10 and 11. The transducers are located in position of Table 6. The first two modes are the controlled modes and the modes 3–5 are the residual modes. By comparing the values of these two tables, it can be seen that, for the optimal locations, the values of the system controllability for the first two modes are greater than the values of an arbitrarily selected location. Thus, the optimization procedure would seem to be effective. In Figure 5, two actuators are used for active vibration control of the simply supported rectangular plate under consideration. In this figure, we would compare the optimal configurations in Table 8 and use the optimal locations of one actuator activated for the first mode and the other actuator for the second mode. As can be seen, the results based on mathematical procedure used based on GA and represented in Table 8 are more effective than the individual activations of each actuator for each mode.Table 10
The effect of values of controllability for optimal location, whenever the first two modes are controlled.
Values of controllability ofith mode
Number of actuators
1
2
3
4
5
One actuator
0.033
0.053
0.000
0.000
0.000
Two actuators
0.016
0.026
0.000
0.000
0.000Table 11
The effect of values of controllability for random location, whenever the first two modes are controlled.
Values of controllability ofith mode
Number of actuators
1
2
3
4
5
One actuator
0.001
0.004
0.010
0.008
0.032
Two actuators
0.001
0.007
0.018
0.010
0.040Frequency response of the plate controlled by the piezoelectric actuator-sensor patches when the first two modes are controlled, while the next three modes are the residual modes: (a) one actuator-sensor and (b) a pair of actuator-sensors.
(a)
(b)Figure 5
Comparison of the optimal configurations in Table8 and use of the optimal locations of one actuator activated for the first mode and the other actuator for the second mode.


## 5.3. Simulation 2

In the second simulation, the first three lower modes must be controlled and the remaining two modes are residual modes. The transducers should be located at random and optimal positions that are listed in Table9. The frequency responses of the plate, excited by a concentrated force and located at x
=
0.43
m and y
=
0.21
m, for single and double transducer configurations, are presented in Figure 6. As can be seen, the optimization criteria are effective in having more damping effect. In Figure 7, the influences of the number of sensors and actuators are compared in the control of vibration. In this simulation, the sensors and actuators are located in the optimal positions and the first two lower modes must be controlled. It can be concluded that when a pair of actuators and sensors are used the damping effect increases and the amplitude vibration of plate reduces more.Frequency response of the plate controlled by the piezoelectric actuator-sensor. The first three modes are the controlled modes, while the next two modes are the residual modes: (a) one actuator-sensor and (b) two actuator-sensors.
(a)
(b)Figure 7
Comparison of frequency response of the plate controlled by different number of piezoelectric actuator-sensors.


## 6. Conclusions

In the present study, the active vibration control and optimal position of piezoelectric patches attached to a thin plate were analyzed. For deriving the equation of motion and sensor output equation, Hamilton’s principle and Rayleigh-Ritz method were used. In the next step, the optimal positions and orientations of piezoelectric actuators and sensors attached to a rectangular plate were determined based on the concept of spatial controllability/observability and considering residual modes, for reducing the spillover effect. GA was utilized for optimizing the locations and orientations of piezoelectric devices. The results indicated that, by locating piezoelectric sensors and actuators in the optimal positions, the damping effect could increase and the amplitudes of plate vibration were reduced more effectively. Furthermore, in general, the active vibration control was effective in the vibration control of the thin plate.



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*Source: 276714-2014-05-05.xml*