Content uploaded by Marcelo Areias Trindade
Author content
All content in this area was uploaded by Marcelo Areias Trindade
Content may be subject to copyright.
Content uploaded by Marcelo Areias Trindade
Author content
All content in this area was uploaded by Marcelo Areias Trindade
Content may be subject to copyright.
DINCON’2008
7 BrazilianConferenceonDynamics,ControlandApplications
th
May 07-09,2008
FCT -Unesp atPresidente Prudente,SP,Brazil
x(i)
y(i)
z(i)
6
5
4
3
2
1
-4
-2
0
2
4
-3
-2
-1
0
1
2
3
APPLICATIONS OF PIEZOELECTRIC SENSORS AND ACTUATORS FOR
ACTIVE AND PASSIVE VIBRATION CONTROL
Marcelo A. Trindade
1
1
Department of Mechanical Engineering, São Carlos School of Engineering, University of São Paulo, Av. Trabalhador São-Carlense, 400,
São Carlos-SP, 13566-590, Brazil, trindade@sc.usp.br
Abstract: The present article provides a brief review of the
open literature concerning applications of piezoelectric sen-
sors and actuators for active and passive vibration control.
Then, some recent advances on this subject are presented.
In particular, the following topics are discussed in detail: i)
modeling of structures with piezoelectric sensors and actu-
ators, ii) evaluation of the effective electromechanical cou-
pling coefficient, iii) applications for active and passive vi-
bration control.
Keywords: Piezoelectric Materials, Active Vibration Con-
trol, Passive Vibration Control
1. INTRODUCTION
When incorporated into a laminated composite structure,
piezoelectric materials can be used as distributed means for
sensing and/or actuating the structure’s response. This is
achieved thanks to the electromechanical interaction that
occurs in a piezoelectric material for which the applica-
tion of a force or pressure produces electric charge/voltage
(direct piezoelectric effect) and the application of electric
charge/voltage is responded by induced strain (converse
piezoelectric effect). Therefore, the dynamic response of a
structure with incorporated piezoelectric layers/patches can
be both monitored, through the measure of the charge/voltage
induced in the piezoelectric element(s) acting as sensor(s),
and controlled, through the application of an appropriate
charge/voltage to the piezoelectric element(s) acting as ac-
tuator(s). It is clear that to effectively control the structure’s
response using piezoelectric sensors and actuators, one must
be able not only to sense and actuate the structure’s response
but also to evaluate the appropriate charge/voltage to be ap-
plied to the actuator(s) based on the measured charge/voltage
induced in the sensor(s). This is achieved by a control sys-
tem that connects the sensor(s) and actuator(s) and can be
designed to induce the required response to the structure.
Over the past two decades, several research works have
shown that the use of integrated piezoelectric patches act-
ing as sensors and/or actuators allows effective control of
the structure’s vibrations [1–4]. These laminated composite
structures with integrated piezoelectric sensors and actuators
form a class of “smart structures” that has been widely used
for structural vibration control in the recent years. The piezo-
electric layers/patches can be either bonded to the surfaces
of the host structure or embedded into a laminate structure.
Surface-bonded, also known as surface-mounted, actuators
have the advantage of ease in construction, access and main-
tainability but may be subjected to high longitudinal stresses
and contact with surrounding objects that may be detrimental
to normally brittle piezoceramic materials. Embedding the
piezoelectric layers/patches alleviate these problems and also
enables good mechanical and electrical link with the struc-
tural element and gluing materials may be unnecessary. On
the other hand, embedding may lead to complex manufactur-
ing and electrical insulation.
1.1. Piezoelectric extension and shear actuation and
sensing modes
Surface-mounted piezoelectric actuators are normally
poled in the thickness direction so that the application of a
through-thickness electric field forces an elongation or con-
traction of the actuators. If the actuators are well-bonded to
the surface of the host structure, their elongation or contrac-
tion causes a deformation of the host structure. This may be
represented as the application of axial forces on the struc-
ture’s surface at the actuator edges leading to bending mo-
ments applied to the structure’s neutral line. That is why,
surface-mounted actuators are also known as extension or
extension-bending actuators. They been widely used on ac-
tive [1], passive [2] and hybrid active-passive [5] control ap-
plications. Although extension actuators can be very effec-
tive when surface-mounted on the host structure, they are not
very effective when embedded in a laminate structure. This
is because they induce smaller bending moments when they
are close to the structure’s neutral line.
Sun and Zhang [6] proposed the use of the thickness-
shear mode of piezoelectric actuators embedded in a sand-
wich beam. In this case, the piezoelectric patches are poled
in the axial direction and, when subjected to the standard
through-thickness electric field, induce thickness shear stress
in the sandwich structure’s core. These piezoelectric ac-
tuators are known as shear actuators and are produced by
some piezoceramic manufacturers normally in the form of
plates poled in the length or width direction. Benjeddou,
Trindade, and Ohayon [7–9] showed that shear actuators in-
duce distributed actuation moments in the structure unlike
extension actuators which induce boundary forces. There-
fore, shear actuation mechanism may lead to less problems
of debonding in actuators boundaries and to minor depen-
dence of the control performance on actuators position and
length. Aldraihem and Khdeir [10–12] presented exact so-
lutions for sandwich beams with shear and extension actua-
tors using equivalent single layer models based on first-order
and third-order shear deformation theories. Trindade, Ben-
jeddou, and Ohayon [13] presented a comparison between
active control performances of shear and extension actuation
mechanisms using a sandwich beam finite element model.
They showed that shear actuators are generally more suit-
able to control bending vibrations of stiff structures. Raja,
Sreedeep, and Prathap [14] have presented a finite element
static analysis of sandwich beams actuated simultaneously
by shear and extension actuators for several boundary con-
ditions. Recent experiments and numerical simulations per-
formed by Baillargeon and Vel [15] have shown that shear
actuators can provide significant reduction on the vibrations
of a sandwich beam. Vel and Batra [16] presented an ex-
act 3D solution for the static cylindrical bending of simply
supported laminated plates with embedded shear piezoelec-
tric actuators. Edery-Azulay and Abramovich [17] also pre-
sented closed-form solutions for the static analysis of lami-
nate/sandwich beams with embedded extension and shear ac-
tuators. It has been observed that extension actuators are gen-
erally more effective for very flexible host structures while
shear actuators are more effective for stiffer structures (e.g.
short beams). Also, the effectiveness of extension actuators
is more dependent on the position along the beam than shear
actuators. These and other distinctive features of extension
and shear actuators may be exploited to study their simulta-
neous use and to design a combined extension-shear actuated
beam [18]. These and other aspects of the shear actuation
mechanism are reviewed in [19].
1.2. Effective electromechanical coupling coefficient
It is well known that the sensing and actuation perfor-
mances of piezoelectric materials depend highly on the effec-
tive electromechanical coupling provided to the structure to
which they are attached. This coupling itself is known to be
dependent on the intrinsic electromechanical coupling coef-
ficient (EMCC) of the piezoelectric material and on the me-
chanical coupling between the piezoelectric sensor/actuator
and the rest of the structure. In particular, the effective
EMCC of a structure with piezoelectric elements should be
expected to be smaller than the material EMCC of the piezo-
electric material embedded into the structure [20].
The EMCC of a piezoelectric material was first intro-
duced by Mason [21] and can be defined as the square root of
the ratio of electrical energy stored in the volume of a piezo-
electric body to the total mechanical energy supplied to the
body (or vice-versa). The EMCC is an important measure of
the effectiveness of the electromechanical coupling and, thus,
of the effectiveness of a piezoelectric material for a given ap-
plication. Several formulas and methods were proposed to
evaluate numerically or measure experimentally the EMCC
of a piezoelectric material [21–23]. Most of them refer to the
material EMCC, which is only function of the material prop-
erties and normally requires an assumption of homogeneous
deformation throughout the piezoelectric body [24].
Some attempts to evaluate the effective EMCC of a struc-
ture with piezoelectric elements were previously published
in the literature [25–27]. Most of them make use of the for-
mulas proposed by Mason [21] and Ulitko [22], which mea-
sure variations in the resulting electromechanical structure
when the electric boundary conditions are changed. These
formulas are intended to account automatically for nonho-
mogeneous deformations in the piezoelectric elements, cor-
responding to an integration of electromechanical coupling
throughout the volume of piezoelectric bodies. This leads to
a quite simple methodology when applying these formulas to
evaluate the effective EMCC from experimentally measured
quantities. However, when applying these formulas to the-
oretical models, care should be taken as to whether the the-
oretical model accounts properly for the electric boundary
conditions [28].
1.3. Modeling of structures with piezoelectric sensors
and actuators
The first model proposed for structures with piezoelectric
actuators was quite simple considering that an induced strain,
proportional to the voltage applied to the actuator, was ap-
plied to the host structure [29]. Improvements of this model
were then proposed by Crawley and de Luis [30], accounting
also for the position of the actuator along the thickness direc-
tion and the shear lag due to the adhesive layer. A somewhat
more sophisticated model based on a Bernoulli-Euler beam
theory was presented in [31]. Herman Shen [32] presented
a Timoshenko beam model to account for the shear strains
induced by the piezoelectric actuator. Lee [33] proposed a
classical laminate theory, that is Kirchoff plate hypotheses
for an equivalent single layer. From the mid 90s on, several
modeling methodologies were proposed for general laminate
structures with piezoelectric layers/patches bonded to, or em-
bedded in, the host structure [34, 35]. Several finite element
models with and without electric degrees of freedom were
also proposed in the literature [36].
1.4. Passive vibration control using shunted piezoelec-
tric materials
The idea of connecting piezoelectric patches to shunt cir-
cuits is basically to control the mechanical energy via the
electrical energy induced in the shunt circuit due to elec-
tromechanical coupling in the piezoelectric [37, 38]. Most
of the recent studies focus on optimizing the shunt circuits
by including resistances, inductances and capacitances in se-
ries and/or parallel [3, 39, 40]. Nevertheless, few studies
focus on the optimization of the electromechanical coupling
in the piezoelectric material. It has been shown that piezo-
electric actuators using their thickness-shear mode can be
more effective than surface-mounted extension piezoelectric
actuators for vibration damping [13, 15, 18, 41]. However,
their use in connection to shunt circuits to provide passive
vibration control is much less explored. In particular, it was
shown that the use of piezoelectric patches in thickness-shear
mode may be more interesting since the electromechanical
coupling is higher than that in extension mode [19, 42–44].
1.5. Active-passive vibration control using piezoelectric
materials
In the last two decades, research was redirected to com-
bined active and passive vibration control techniques [45].
One of these techniques, so-called Active-Passive Piezoelec-
tric Networks (APPN), integrates an active voltage source
with a passive resistance-inductance shunt circuit to a piezo-
electric sensor/actuator [46]. This technique allows to simul-
taneously dissipate passively vibratory energy through the
shunt circuit and actively control the structural vibrations. It
has been shown that combined active-passive vibration con-
trol allows better performance with smaller cost than separate
active and passive control, provided the simultaneous action
is optimized. On the other hand, it has been shown in previ-
ous studies [18, 47] that both purely active and purely passive
damping performance can be improved by properly selecting
between, or combining, extension and thickness-shear actu-
ation mechanisms of piezoelectric actuators/sensors. Hence,
it is expected that the choice between actuation mechanisms
should be important as well for active-passive vibration con-
trol.
2. MODELING OF STRUCTURES WITH PIEZO-
ELECTRIC MATERIALS
In this section, a general methodology for the variational
formulation of coupled equations of motion for structures
with piezoelectric materials is presented. Equations are writ-
ten in terms of both electric potential and electric charge in
the piezoelectric elements. Equipotentiality over each piezo-
electric element electrodes is accounted for in both formula-
tions. Finally, a methodology for coupling the piezoelectric
elements with electric circuits is presented.
2.1. Electric potential formulation
First, a formulation considering electric potential in the
piezoelectric elements as variables is proposed. For that,
let us start by denoting the generalized displacements of the
structure as
¯
u. Hence, the virtual work done by inertial and
external forces can be written as
δT = −
Z
Ω
δ
¯
u
t
ρ
¨
¯
u dΩ (1)
δW =
Z
Ω
δ
¯
u
t
f dΩ (2)
where
¨
¯
u and f stand for the generalized accelerations and ap-
plied body forces vectors, respectively. ρ is the volumetric
mass density and Ω is the volume of the structure. The vir-
tual work done by internal forces can be found from the vir-
tual variation of the electromechanic potential energy. In this
first formulation, it is chosen to write the potential energy
as the electric Gibbs energy, written in terms of mechanical
strains ε and electric fields E, such that its variation reads
δU(ε, E) =
Z
Ω
δε
t
c
E
ε − δε
t
eE − δE
t
e
t
ε − δE
t
ε
E
dΩ
(3)
where c
E
, e and
ε
are the matrices of elastic (for constant
electric field), piezoelectric and dielectric (for constant me-
chanical strain) constants of the material.
These virtual work expressions can be spatially dis-
cretized through the discretization of the displacements
fields, and thus of the corresponding strains, such that
¯
u(x
1
,x
2
,x
3
,t) = N
u
(x
1
,x
2
,x
3
)u(t) (4)
ε(x
1
,x
2
,x
3
,t) = B(x
1
,x
2
,x
3
)u(t) (5)
The electric fields appearing in (3) can also be discretized
and written in terms of difference of electric potential (volt-
age) as
E(x
1
,x
2
,x
3
,t) = N
V
(x
1
,x
2
,x
3
)V(t) (6)
Then, replacing discretized fields into the virtual work ex-
pressions yields
δT = −δu
t
M
¨
u ; M =
Z
Ω
ρN
t
u
N
u
dΩ (7)
δW = δu
t
F
m
; F
m
=
Z
Ω
N
t
u
f dΩ (8)
δU = δu
t
K
E
u
u − δu
t
K
uv
V − δV
t
K
t
uv
u − δV
t
K
v
V (9)
where the elastic (for constant electric field), piezoelectric
and dielectric (for constant mechanical strain) stiffness ma-
trices are
K
E
u
=
Z
Ω
B
t
c
E
B dΩ ; K
uv
=
Z
Ω
B
t
eN
V
dΩ
K
v
=
Z
Ω
N
t
V
ε
N
V
dΩ
(10)
The equations of motion can then be derived from
D’Alembert’s principle,
δT −δU +δW = 0 (11)
such that
δu
t
(M
s
+ M
p
)
¨
u + (K
us
+ K
E
up
)u − K
uv
V − F
m
+δV
t
−K
t
uv
u − K
v
V
= 0 (12)
or in matrix form
M
s
+ M
p
0
0 0
¨
u
¨
V
+
K
us
+ K
E
up
−K
uv
−K
t
uv
−K
v
u
V
=
F
m
0
(13)
where M
s
and K
us
are the mass and elastic stiffness matrices
of the structure (without piezoelectric elements) and M
p
and
K
E
up
are the mass and elastic (for constant electric fields) stiff-
ness matrices of the piezoelectric elements. K
uv
and K
v
are
the piezoelectric and dielectric stiffnesses of the piezoelec-
tric elements. F
m
is a vector of the mechanical loads applied
to the structure. The degrees of freedom (dofs) u are the gen-
eralized displacements and V are the generalized differences
of electric potentials (voltages) on the piezoelectric material.
To account for the equipotential condition on the elec-
trodes of each piezoelectric element, let us define the vectors
of differences of electric potentials V
p
induced or applied to
the electrodes of the piezoelectric elements, such that
V = L
p
V
p
(14)
The boolean matrix L
p
has dimension N ×N
p
, where N is
the number of spatial (nodal) points and N
p
is the number of
independent piezoelectric elements. L
p
allows to set an equal
value to selected nodal differences of electric potentials.
Substituting equation (14) in equation (13) and pre-
multiplying the second line of the resulting equation by L
t
p
leads to
M
s
+ M
p
0
0 0
¨
u
¨
V
p
+
K
us
+ K
E
up
−
¯
K
uv
−
¯
K
t
uv
−
¯
K
v
u
V
p
=
F
m
0
(15)
where
¯
K
uv
= K
uv
L
p
;
¯
K
v
= L
t
p
K
v
L
p
(16)
Now, it is worthwhile to investigate three separate cases.
If the piezoelectric elements are short-circuited, the differ-
ences of electric potentials between their electrodes vanish
and, hence, the first line of (15) reduces to
(M
s
+ M
p
)
¨
u + (K
us
+ K
E
up
)u = F
m
(17)
On the other hand, if the piezoelectric elements are open-
circuited, the differences of electric potentials between their
electrodes are unknown and can be evaluated using the sec-
ond line of equation (15), such that
V
p
= −
¯
K
−1
v
¯
K
t
uv
u (18)
Replacing equation (18) in the first line of equation (15) leads
to the following condensed equations of motion
(M
s
+ M
p
)
¨
u +
K
us
+ (K
E
up
+
¯
K
uv
¯
K
−1
v
¯
K
t
uv
)
u = F
m
(19)
From equation (19), the generalized displacements u can
be evaluated. Then, the electric potentials V
p
induced in the
piezoelectric elements can be found using equation (18). It
is worthwhile to notice, from equation (19), that the induced
potentials in the sensors due to the direct piezoelectric effect
lead to an increase in their constant electric field stiffnesses.
This is due to an equivalent electric load generated in the
piezoelectric layer by the induced potential.
Finally, if differences of electric potential are applied to
the piezoelectric elements, the second equation in (15) is au-
tomatically satisfied, since δV
p
= 0, and the first equation in
(15) can be written as
(M
s
+ M
p
)
¨
u + (K
us
+ K
E
up
)u = F
m
+ F
p
(20)
where
F
p
=
¯
K
uv
V
p
(21)
In this case, the piezoelectric elements act as actuators
applying piezoelectric equivalent forces on the structure and
their stiffnesses are those for constant electric field.
2.2. Electric charge formulation
An electric charge formulation can be obtained by using
the Helmholz free energy, written in terms of mechanical
strains ε and electric displacements D, as potential energy
instead of the electric Gibbs energy, such that the virtual vari-
ation of the potential energy is
δU(ε, D) =
Z
Ω
δε
t
c
D
ε − δε
t
hD − δD
t
h
t
ε − δD
t
β
ε
D
dΩ
(22)
where c
D
, h and β
ε
are the matrices of elastic (for constant
electric displacement), piezoelectric and dielectric (for con-
stant mechanical strain) constants of the material.
In this case, the discretization of mechanical quantities
(displacements and strains) is the same as in the previous
section. The difference is on electrical quantities, that is,
the electric displacements appearing in (22), which are dis-
cretized as
D = N
D
D
n
(23)
Hence, the discretized version of the potential energy is
written as
δU = δu
t
K
D
u
u − δu
t
K
ud
D
n
−δD
t
n
K
t
ud
u + δD
t
n
K
d
D
n
(24)
where the elastic (for constant electric displacement), piezo-
electric and dielectric stiffness matrices are
K
D
u
=
Z
Ω
B
t
c
D
B dΩ ; K
ud
=
Z
Ω
B
t
hN
D
dΩ
K
d
=
Z
Ω
N
t
D
β
ε
N
D
dΩ
(25)
Again, from D’Alembert’s principle (11), the equations
of motion are now written in terms of the generalized dis-
placements u and electric displacements D
n
, such that
δu
t
(M
s
+ M
p
)
¨
u + (K
us
+ K
D
up
)u − K
ud
D
n
− F
m
+δD
t
n
−K
t
ud
u + K
d
D
n
= 0 (26)
or in matrix form
M
s
+ M
p
0
0 0
¨
u
¨
D
n
+
K
us
+ K
D
up
−K
ud
−K
t
ud
K
d
u
D
n
=
F
m
0
(27)
where, as in the previous case, M
s
and K
us
are the mass
and elastic stiffness matrices of the structure (without piezo-
electric elements) and M
p
and K
D
up
are the mass and elastic
(for constant electric displacements) stiffness matrices of the
piezoelectric elements. K
ud
and K
d
are the piezoelectric and
dielectric stiffnesses of the piezoelectric elements.
To account for the equipotential condition on the elec-
trodes of each piezoelectric element, let us define the vectors
of electric charges q
p
on the electrodes of the piezoelectric
elements, such that
D
n
= B
p
q
p
; B
p
= L
p
A
−1
p
(28)
The boolean matrix L
p
has dimension N × N
p
, where N
is the number of spatial (nodal) points and N
p
is the number
of independent piezoelectric elements. L
p
allows to set an
equal value to selected nodal electric displacements. A
p
is a
diagonal matrix with the surface area of the electrodes of the
piezoelectric elements.
Substituting equation (28) in equation (27) and pre-
multiplying the second line of the resulting equation by B
t
p
leads to
M
s
+ M
p
0
0 0
¨
u
¨
q
p
+
K
us
+ K
D
up
−K
uq
−K
t
uq
K
q
u
q
p
=
F
m
0
(29)
where
K
uq
= K
ud
B
p
; K
q
= B
t
p
K
d
B
p
(30)
Now, it is worthwhile to investigate two separate cases.
If the piezoelectric elements are open-circuited, the electric
charge flux between their electrodes vanish and, hence, the
first line of (29) reduces to
(M
s
+ M
p
)
¨
u + (K
us
+ K
D
up
)u = F
m
(31)
On the other hand, if the piezoelectric elements are short-
circuited, the electric charge flux between their electrodes are
unknown and can be evaluated using the second line of equa-
tion (29), such that
q
p
= K
−1
q
K
t
uq
u (32)
Replacing equation (32) in the first line of equation (29) leads
to the following condensed equations of motion
(M
s
+ M
p
)
¨
u +
K
us
+ (K
D
up
− K
uq
K
−1
q
K
t
uq
)
u = F
m
(33)
From equation (33), the generalized displacements u can
be evaluated. Then, the electric charges q
p
flowing between
electrodes of the piezoelectric elements can be found using
equation (32). It is worthwhile to notice, from equation (33),
that the induced electric displacements in the sensors due to
the direct piezoelectric effect lead to a decrease in their stiff-
nesses. This is due to the relaxation of the equivalent electric
load generated in the piezoelectric layer by the induced po-
tential.
Finally, if electric charges are applied to the piezoelectric
elements, the second equation in (29) is automatically sat-
isfied, since δq
p
= 0, and the first equation in (29) can be
written as
(M
s
+ M
p
)
¨
u + (K
us
+ K
D
up
)u = F
m
+ F
p
(34)
where
F
p
= K
uq
q
p
(35)
in this case, the piezoelectric elements act as charge actuators
applying piezoelectric equivalent forces on the structure and
their stiffnesses are those for constant electric displacement.
2.3. Connection to electric circuits
It is worthwhile to analyze the connection of piezoelec-
tric elements to electric circuits, specially when shunt circuits
are considered for passive vibration control. To this end, it
seems that an electric charge formulation is more appropri-
ate since it is possible to relate the electric charges flowing
between the piezoelectric elements electrodes with the elec-
tric charges flowing through the electric circuit. First, let us
consider a set of simple but quite general electric circuits
composed of an inductor, a resistor and a voltage source.
The equations of motion for such circuits can be found us-
ing d’Alembert’s principle, such that the virtual work done
by the inductors δT
L j
, resistors, δW
R j
, and voltage sources,
δW
V j
, of the j-th electric circuit are
δT
L j
= −δq
c j
L
c j
¨q
c j
; δW
R j
= −δq
c j
R
c j
˙q
c j
;
δW
V j
= δq
c j
V
c j
(36)
where L
c j
, R
c j
e V
c j
are the inductance, resistance and ap-
plied voltage of the j-th electric circuit. q
c j
is the electric
charge flowing through the j-th electric circuit. Combining
the virtual work done by all circuits leads to
δT
L
=
n
∑
j=1
δT
L j
= −δq
t
c
L
c
¨
q
c
δW
R
=
n
∑
j=1
δW
R j
= −δq
t
c
R
c
˙
q
c
δW
V
=
n
∑
j=1
δW
V j
= δq
t
c
V
c
(37)
where q
c
is the vector of electric charges, L
c
and R
c
are di-
agonal matrices with the inductances and resistances of each
circuit, and V
c
is the vector of applied voltages.
Adding these virtual works to the electromechanical vir-
tual works of previous section, such that
δT −δU +δW + δT
L
+δW
R
+δW
V
= 0 (38)
or, in terms of the generalized displacements,
δu
t
(M
s
+ M
p
)
¨
u + (K
us
+ K
D
up
)u − K
uq
q
p
− F
m
+δq
t
p
(−K
t
uq
u + K
q
q
p
) + δq
t
c
(L
c
¨
q
c
+ R
c
˙
q
c
− V
c
) = 0
(39)
Then, the connection between each piezoelectric element
and a corresponding electric circuit is done by stating that the
electric charges flowing from the piezoelectric element enter
the circuit and vice-versa, such that
q
c
= q
p
(40)
Thus, replacing q
c
by q
p
in (39) leads to the following
coupled equations of motion
M
s
+ M
p
0
0 L
c
¨
u
¨
q
p
+
0 0
0 R
c
˙
u
˙
q
p
+
K
us
+ K
D
up
−K
uq
−K
t
uq
K
q
u
q
p
=
F
m
V
c
(41)
In this case, the solution for u and q
p
must be simulta-
neous, that is accounting for the electromechanical and cir-
cuit equations of motion. Notice that the passive components
of the electric circuit L
c
and R
c
affect the equivalent piezo-
electric force applied to the structure when an actuator with
applied voltage is considered. For a simple actuator with ap-
plied voltage, that is with only a voltage source in the circuit
(L
c
= R
c
= 0), the second equation in (41) can be solved for
q
p
leading to
q
p
= K
−1
q
V
c
+ K
−1
q
K
t
uq
u (42)
which can be substituted in (41) such that it reduces to
(M
s
+ M
p
)
¨
u +
K
us
+ (K
D
up
− K
uq
K
−1
q
K
t
uq
)
u = F
m
+ F
p
(43)
where the equivalent piezoelectric force F
p
applied to the
structure by the piezoelectric actuators is
F
p
= K
uq
K
−1
q
V
c
(44)
From equation (43), the generalized displacements u in-
duced by mechanical and piezoelectric equivalent forces can
be evaluated. Then, the electric charges q
p
flowing between
electrodes of the piezoelectric elements can be found using
equation (42).
3. EVALUATION OF THE EFFECTIVE ELEC-
TROMECHANICAL COUPLING COEFFICIENT
3.1. Evaluation of material EMCC
Probably the most interesting method to evaluate the ma-
terial EMCC is based on a closed-loop quasi-static energy
cycle [48] so that the effective energy conversion from me-
chanical to electrical and vice-versa can be computed. One
possible energy cycle is shown in Figure 1. It is obtained
by first mechanically loading the piezoelectric body in open-
circuit condition (Figure 1, a → b, initial (a: dashed) and
final (b: solid) configurations) so that both mechanical and
electrical energies U
oc
are stored in the piezoelectric body.
Then, the surfaces with electrodes are held to constrain the
deformation of the piezoelectric body and an ideal electric
load is connected to the electrodes as shown in Figure 1
(b → c). In this step, part of the energy stored in the body
is converted into work U
conv
done in the electric load. The
electric load could be for instance a resistance so that the
electrical energy would be converted into heat and dissipated
in the surrounding environment. When all electrical energy is
dissipated the difference of potential between the piezoelec-
tric body electrodes should be zero. Finally, the piezoelectric
body is short-circuited and released from its holders so that
it deforms back to its original configuration, performing a
purely mechanical work U
sc
. In Figure 1, U
oc
, U
conv
and U
sc
are the areas of the triangles [abd], [abc] and [acd], respec-
tively. From this energy cycle, it is possible to define the
square EMCC as the following energy conversion efficiency
k
2
i j
=
U
conv
U
oc
=
U
oc
−U
sc
U
oc
(45)
Notice that a similar energy conversion efficiency analysis
could be performed with other energy cycles, for instance,
through electrical loading and measurement of energy ratio
converted into mechanical work. Notice also that the EMCC
indices i and j state for the electric field/displacement and
strain/stress components, respectively.
D
3
= 0
E
3
= 0
U
sc
U
conv
σ
3
ε
3
ε
b
a
b
c
a → b
b → c
c → a
+
–
+
–
E
3
i
U
oc
d
Figure 1 – Energy cycle for a piezoelectric body to evaluate elec-
tromechanical conversion energy.
Let us now consider equation (45) to derive the EMCC in
terms of material properties. For linear and unidimensional
(direction 3) piezoelectric constitutive equations, the internal
energy stored in the piezoelectric body can be written as
U =
1
2
Z
Ω
(σ
3
ε
3
+ E
3
D
3
) dΩ (46)
Using the e-form constitutive equations, the mechanical
stress σ
3
and electric displacement D
3
can be expressed in
terms of the mechanical strain ε
3
and electric field E
3
as
σ
3
= ¯c
E
33
ε
3
− ¯e
33
E
3
D
3
= ¯e
33
ε
3
+
¯
ε
33
E
3
(47)
where ¯c
E
33
, ¯e
33
and
¯
ε
33
are the elastic (at constant electric
field), piezoelectric and dielectric (at constant strain) con-
stants. One can notice that during the first (a → b) and third
(c → a) phases of the energy cycle of Figure 1, that is unidi-
mensional contraction (extension) in direction 3 with D
3
= 0
(E
3
= 0), the constitutive equations (47) can be reduced to:
First phase of energy cycle (a → b)
σ
3
= ¯c
D
33
ε
3
with ¯c
D
33
= ¯c
E
33
+ ( ¯e
2
33
/
¯
ε
33
)
E
3
= −( ¯e
33
/
¯
ε
33
)ε
3
and D
3
= 0
(48)
Third phase of energy cycle (c → a)
σ
3
= ¯c
E
33
ε
3
D
3
= ¯e
33
ε
3
and E
3
= 0
(49)
Hence, the total energy stored in the piezoelectric body in
the first and third phases of the energy cycle can be evaluated
from equations (46), (48) and (49), such that
U
oc
=
1
2
Z
Ω
¯c
D
33
ε
2
3
dΩ (50)
U
sc
=
1
2
Z
Ω
¯c
E
33
ε
2
3
dΩ (51)
Considering homogeneous deformation throughout the
piezoelectric body, the square EMCC for this mode of de-
formation can be written from equations (45), (50) and (51)
as
k
2
33
=
¯c
D
33
− ¯c
E
33
¯c
D
33
(52)
which, from equation (48), can be simplified to
k
2
33
=
¯e
2
33
¯c
D
33
¯
ε
33
=
¯e
2
33
¯c
E
33
¯
σ
33
(53)
A similar expression can be obtained using the formula
proposed in [49] and based on the ratio between the square of
a so-called mutual elasto-dielectric energy U
m
and the prod-
uct of the stored elastic U
e
and dielectric U
d
energies, such
that
k
2
33
=
U
2
m
U
e
U
d
(54)
where the mutual, elastic and dielectric energies should be
defined as
U
m
=
1
2
Z
Ω
¯
d
33
σ
3
E
3
dΩ ; U
e
=
1
2
Z
Ω
¯s
E
33
σ
2
3
dΩ ;
U
d
=
1
2
Z
Ω
¯
σ
33
E
2
3
dΩ
(55)
Although the energy cycle (Figure 1) and the EMCC def-
inition (Equation (45)) are defined for quasi-static defor-
mation, this analysis may be extended to resonant vibra-
tions. However, in such cases, the deformation throughout
the piezoelectric body may not be homogeneous and, thus,
both open-circuit and short-circuit energies must be evalu-
ated through integration over the volume of the piezoelectric
body. Supposing that the displacement u
3
and, consequently
the mechanical strain ε
3
, along the x
3
direction are written as
u
3
(x
3
,t) = φ(x
3
)cosωt ; ε
3
(x
3
,t) = φ
0
(x
3
)cosωt, (56)
the strain, as in (50) and (51), and kinetic energies of the
piezoelectric body are
U =
1
2
cos
2
ωt
Z
Ω
¯c
33
φ
0
(x
3
)
2
dΩ ;
T =
1
2
ω
2
sin
2
ωt
Z
Ω
ρφ(x
3
)
2
dΩ
(57)
Recalling that, for resonant vibrations, the maximum ki-
netic energy equals the maximum strain energy, and sup-
posing that the vibration mode φ(x
3
) remains unchanged
for open-circuit and short-circuit conditions, the maximum
strain energy in these electric conditions reads
U
max
oc
=
1
2
Z
Ω
¯c
D
33
φ
0
(x
3
)
2
dΩ =
1
2
ω
2
oc
¯
T ;
U
max
sc
=
1
2
Z
Ω
¯c
E
33
φ
0
(x
3
)
2
dΩ =
1
2
ω
2
sc
¯
T
(58)
where
¯
T =
Z
Ω
ρφ(x
3
)
2
dΩ (59)
Notice that the integrals in U
max
oc
and U
max
sc
can be seen
as the modal stiffness in open and short circuit conditions
whereas
¯
T is the modal mass. The square EMCC, for this
particular vibration mode, can then be defined as
k
2
33
=
U
max
oc
−U
max
sc
U
max
oc
=
ω
2
oc
−ω
2
sc
ω
2
oc
(60)
For homogeneous properties throughout the piezoelectric
body, equation (60) reduces to equation (45). This analysis,
however, is only valid under the assumption that the electric
field E
3
and displacement D
3
can be set to zero throughout
the piezoelectric body in short-circuit and open-circuit con-
ditions, respectively.
3.2. Evaluation of effective EMCC
It is also worthwhile to evaluate the effective EMCC for
a general mechanical structure with bonded and/or embed-
ded piezoelectric elements. In this case, the effective EMCC
should account not only for the material EMCC of each
piezoelectric element but also for the mechanical coupling
between the piezoelectric elements and the rest of the struc-
ture. Indeed, a satisfactory measure of the effective square
EMCC for a structure with piezoelectric elements should be
the ratio of electrical energy stored in the piezoelectric ele-
ments to the total mechanical (strain) energy supplied to the
structure. However, the evaluation of such effective EMCC
requires accounting for non-homogeneous properties and de-
formation. Hence, a general procedure based on a model of
the structure with piezoelectric elements is proposed.
Starting from the reduced equations of motion (17) and
(19), the i-th eigenmode and eigenfrequency of the structure
with piezoelectric elements in short-circuit and open-circuit
can be evaluated, respectively, by
h
−ω
i
sc
2
M + (K
s
+ K
sc
p
)
i
T
i
sc
= 0 (61)
h
−ω
i
oc
2
M + (K
s
+ K
oc
p
)
i
T
i
oc
= 0 (62)
where, for the sake of clarity, the matrices M, K
s
, K
sc
p
and
K
oc
p
are defined as
M = M
s
+ M
p
; K
s
= K
us
; K
sc
p
= K
E
up
;
K
oc
p
= K
E
up
+
¯
K
uv
¯
K
−1
v
¯
K
t
uv
(63)
Supposing that the short-circuit and open-circuit condi-
tions of the piezoelectric elements do not yield large varia-
tions in the structure eigenmodes (T
i
sc
= T
i
oc
= T
i
), which
should be valid for relatively small piezoelectric elements,
and for mass-normalized eigenmodes, the i-th short-circuit
and open-circuit eigenfrequencies can be written as
ω
i
sc
2
= T
t
i
(K
s
+ K
sc
p
)T
i
(64)
and
ω
i
oc
2
= T
t
i
(K
s
+ K
oc
p
)T
i
(65)
Therefore, the effective square EMCC for the structure
with piezoelectric elements, vibrating in the i-th mode, can
be defined as
K
2
i
=
ω
i
oc
2
−ω
i
sc
2
ω
i
oc
2
=
T
t
i
(K
oc
p
− K
sc
p
)T
i
T
t
i
(K
s
+ K
oc
p
)T
i
(66)
Equation (66) provides a relatively simple technique to
evaluate the effective EMCC in terms of the open-circuit and
short-circuit eigenfrequencies, which may be measured for
a given experimental setup or numerically calculated for a
given structural model. Notice that this equation provides an
average effective EMCC for a structure with several piezo-
electric elements and may even account for the electrome-
chanical coupling provided by a set of piezoelectric elements
working in different deformation modes (e.g. 33, 31, 15).
Hence, if the objective is to evaluate the effective EMCC
provided by a single piezoelectric element, this evaluation
should be done by imposing a short-circuit condition for the
other piezoelectric elements, so that the only electrically-
stiffened element will be the element under study.
Notice that, for the analysis of a single piezoelectric el-
ement with homogeneous properties and working mainly in
one deformation mode, K
sc
p
≈ (1 − k
2
jl
)K
oc
p
, where k
2
jl
is the
material square EMCC for the specific deformation mode.
Thus, in this case, K
2
i
may be approximated as
¯
K
2
i
= k
2
jl
T
t
i
K
oc
p
T
i
T
t
i
(K
s
+ K
oc
p
)T
i
(67)
This expression can be interpreted as the product of the
material square EMCC by the ratio of the OC strain en-
ergy stored in the piezoelectric element to the total strain
energy. This formula is similar to that in the Modal Strain
Energy method, proposed in [50] for the evaluation of ef-
fective modal loss factors for structures with viscoelastic el-
ements. An analysis of equation (67) indicates that there
are two main levers to maximize the effective EMCC: i) the
material EMCC for the main deformation mode, and ii) the
energy ratio stored in the piezoelectric element for a given
structural eigenmode.
3.3. Comparison with experimental results for a can-
tilever beam
A comparative analysis is performed in this subsection
for an aluminum cantilever beam with two thickness-poled
piezoelectric patches bonded symmetrically on its top and
bottom surfaces near the clamp (Figure 2). The beam has
a length of 243.5 mm, a thickness of 2 mm and a width
of 30 mm, and its material properties are: Young modu-
lus 69 GPa, mass density 2790 kg m
−3
and Poisson ra-
tio 0.3. The piezoelectric patches are identical and made
of a PIC255 piezoceramic with thickness 0.25 mm, width
20 mm, length 25 mm, mass density 7800 kg m
−3
, equiv-
alent SC Young modulus 62.1 GPa, equivalent piezoelec-
tric coefficient e
31
= 11.2 C m
−2
and dielectric coefficient
σ
33
= 15.5 nF m
−1
. These data were adapted from [51]. No-
tice, however, that the piezoceramics width was considered
equal to that of the beam (30 mm) in the calculations using
the present beam model.
25
2.00
0.25
PIC255 Piezoceramic
Aluminum
17
243.50
PIC255 Piezoceramic
0.25
20
30
Figure 2 – Representation of the cantilever beam with bonded
piezoelectric patches (dimensions in mm).
Experimental measurements of the two first bending
eigenfrequencies of the cantilever beam with short-circuit
and open-circuit piezoelectric patches were presented in [51].
The authors then used the measured eigenfrequencies to eval-
uate the effective EMCC and predicted shunted damping ra-
tios. The short-circuit and open-circuit eigenfrequencies and
the resulting EMCC, taken from [51], are presented in Ta-
ble 1. The corresponding results obtained with the present
FE model, accounting for the equipotentiality condition for
both piezoelectric patches, are also presented. It is possible
to observe that the numerical eigenfrequencies match quite
well with the experimental ones (up to 10% error for the first
eigenfrequency). Possible reasons for the error could be the
lack of precise knowledge on the material properties and the
difference between theoretical and real clamped-free bound-
ary conditions. As a result, the numerically evaluated EMCC
do not match exactly the experimental ones. However, it is
possible to observe from Table 1 that both numerical and
experimental results indicate a higher EMCC for the first
eigenmode, as expected, since the piezoelectric patches were
bonded near the clamped end where the normal strains are
higher for the first eigenmode. It is also noticeable that 2/3
of the evaluated square EMCC, to adjust for the difference
of piezoelectric volume, matches quite well the experimental
values. As for the previous case, equation (67) provides a
good approximation for the EMCC.
3.4. Evaluation of equipotential effect on EMCC
0 100 200
0
2
4
PZT length (mm)
Square EMCC M1 (%)
0 100 200
0
2
4
PZT length (mm)
Square EMCC M2 (%)
0 100 200
0
2
4
PZT length (mm)
Square EMCC M3 (%)
0 100 200
0
2
4
PZT length (mm)
Square EMCC M4 (%)
0 100 200
0
2
4
PZT length (mm)
Square EMCC M5 (%)
0 100 200
0
2
4
PZT length (mm)
Square EMCC M6 (%)
0 100 200
0
2
4
PZT length (mm)
Square EMCC M7 (%)
0 100 200
0
2
4
PZT length (mm)
Square EMCC M8 (%)
0 100 200
0
2
4
PZT length (mm)
Square EMCC M9 (%)
Figure 3 – Effect of piezoelectric patch length on EMCC using
models without (dashed) and with (solid) equipotential.
Using the previous case as a baseline configuration (Fig-
ure 2), an analysis of the effect of charge cancelation due
to equipotentiality on the electrodes of a piezoelectric patch
is conducted. This is done by increasing the length of the
piezoelectric patch in the range [10-200] mm and evaluating
the effective EMCC with the models with and without the
equipotentiality condition. Figure 3 shows the square EMCC
for the first nine bending modes. It can be observed that the
model without the equipotentiality condition predicts a con-
tinuous increase of the EMCC with augmenting patch length
(dashed line in Figure 3). This is in accordance with the no-
tion that the amount of energy ratio in the piezoelectric patch
increases and, since its material EMCC is constant (accord-
ing to this model), the ratio of converted energy should in-
crease. However, this is not observed in reality since there
should be an electric charge flux in the longitudinal direc-
tion due to the electrodes equalizing the electric potential
over the piezoelectric patch surface. This effect is known as
charge cancelation. The model accounting for the equipo-
tentiality, on the other hand, does allow the redistribution
of the electric charge so that electric potential is constant
along the longitudinal direction. Therefore, the open cir-
cuit condition does not mean total absence of electric dis-
placement and, thus, the difference between closed and open
circuit could be smaller than the one predicted by the first
model (without equipotentiality). As it can be observed in
Figure 3, the model with equipotentiality condition (solid
line) predicts smaller values for the effective EMCC as the
patch length increases. For very short piezoelectric patches,
the difference between the two models is not very notice-
able on the first bending modes. However, in all cases, the
maximum attainable EMCC is overestimated by the model
without the equipotentiality condition. Notice that although
200 mm long piezoelectric patches are not easily available
commercially, the analysis is still valid since the same effect
should appear if several smaller piezoelectric patches dis-
tributed along the beam were to be connected to the same
electrode. Figure 3 also shows that the charge cancelation is
higher for higher modes so that the effective modal EMCC is
decreasing with modes when equipotentiality is considered
rather than increasing when it is not considered.
0 10 20
0
2
4
PZT segments
Square EMCC M1 (%)
0 10 20
0
2
4
PZT segments
Square EMCC M2 (%)
0 10 20
0
2
4
PZT segments
Square EMCC M3 (%)
0 10 20
0
2
4
PZT segments
Square EMCC M4 (%)
0 10 20
0
2
4
PZT segments
Square EMCC M5 (%)
0 10 20
0
2
4
PZT segments
Square EMCC M6 (%)
0 10 20
0
2
4
PZT segments
Square EMCC M7 (%)
0 10 20
0
2
4
PZT segments
Square EMCC M8 (%)
0 10 20
0
2
4
PZT segments
Square EMCC M9 (%)
Figure 4 – Effect of piezoelectric patch segmentation on EMCC
using models without (dashed) and with (solid) equipotential.
The previous analysis indicates that smaller and indepen-
dent piezoelectric patches could be more interesting for en-
ergy conversion. Hence, in an attempt to better understand
this effect, a comparison between the energy converted by a
single electroded piezoelectric patch (200 mm long starting
10 mm from the clamped end) and by the same amount of
piezoelectric material but segmented in smaller patches with
independent electrodes was performed. A 1 mm long sepa-
ration is considered between any two segments. The results
were evaluated using the two models (with and without the
equipotentiality condition) and are shown in Figure 4. It can
be noticed that the optimal number of piezoelectric segments
depends on the mode number, but it seems that the higher the
mode number the higher the number of segments that should
be used for an optimal EMCC. For instance, for the first
mode the maximum EMCC is obtained with two 100 mm
long independent patches whereas for modes 6 to 9, twenty
patches of 10 mm each (or more and shorter) should be used
(solid lines in Figure 4). This is accordance with the previous
analysis, since higher segmentation leads to smaller piezo-
Table 1 – Numerical and experimental eigenfrequencies and square EMCC for the cantilever beam.
Numerical Experimental
f
oc
(Hz) f
sc
(Hz) K
2
i
(%)
¯
K
2
i
(%) (2/3)K
2
i
(%) f
oc
(Hz) f
sc
(Hz) K
2
i
(%)
1 29.31 29.15 1.11 1.11 0.74 26.64 26.56 0.60
2 177.41 177.10 0.36 0.38 0.25 165.76 165.63 0.18
electric patches and, thus, to less charge cancelation. Notice
also that the higher the number of segments, i.e. the smaller
the patch length, the smaller is the difference between the
two models since the effect of charge cancelation is smaller.
This effect was also suggested more than sixty years ago by
Cady [52] through the use of independent short electrodes on
a single piezoelectric patch.
3.5. Effective shear EMCC assessment
In this subsection, a parametric analysis is performed for a
sandwich beam with several shear piezoelectric patches em-
bedded in the beam’s core in order to maximize some se-
lected modal effective EMCC. For that, let us consider the
sandwich beam shown in Figure 5. It consists of two alu-
minum faces, for which the properties are: Young modulus
69 GPa and mass density 2690 kg m
−3
, and a rigid foam core,
for which the properties are: Young modulus 62 MPa, shear
modulus 21 MPa and mass density 80 kg m
−3
. Five PIC255
piezoceramic patches are embedded in the sandwich beam
core, replacing part of the rigid foam. The material proper-
ties of PIC255 are: mass density 7800 kg m
−3
, SC Young
modulus 62.1 GPa, SC shear modulus 24 GPa, piezoelec-
tric coefficient e
15
= 13.7 C m
−2
and dielectric coefficient
σ
11
= 15.5 nF m
−1
. The width of all layers was considered
to be equal to 25 mm.
L
d a e a e a e a e a
H
H
h
Aluminum
Aluminum
PZT
Rigid Foam
PZT PZT PZT
PZT
Figure 5 – Sandwich beam with several embedded shear piezo-
electric patches.
First, a parametric analysis of spacing and length of the
piezoelectric patches is performed. For that, other geometric
parameters are fixed at L = 220 mm, d = 10 mm, H = 3 mm
and h = 1 mm. The length of the patches a is set to vary in the
range [5–25] mm and the patches are positioned through the
spacing between patches e, which is set to vary in the range
[5–30] mm (Figure 5). The effective EMCC for each config-
uration is then evaluated using equation (66), for which all
patches are set to either short-circuit or open-circuit condi-
tion at the same time. This means that the effective EMCC is
composed by contributions of each piezoelectric patch elec-
tromechanical coupling. Figure 6 shows the average and
modal square EMCC for the first five bending modes. It in-
dicates that the EMCC is higher for well-spaced short piezo-
electric patches (a = 5 mm and e = 30 mm), although the op-
5
10
15
20
25
30
5
10
15
20
25
0
0.2
0.4
0.6
0.8
Length (mm)
Spacing (mm)
Average Square EMCC (%)
5
10
15
20
25
30
5
10
15
20
25
0
0.2
0.4
0.6
0.8
Length (mm)
Spacing (mm)
Square EMCC M1 (%)
5
10
15
20
25
30
5
10
15
20
25
0
0.2
0.4
0.6
0.8
Length (mm)
Spacing (mm)
Square EMCC M2 (%)
5
10
15
20
25
30
5
10
15
20
25
0
0.2
0.4
0.6
0.8
Length (mm)
Spacing (mm)
Square EMCC M3 (%)
5
10
15
20
25
30
5
10
15
20
25
0
0.2
0.4
0.6
0.8
Length (mm)
Spacing (mm)
Square EMCC M4 (%)
5
10
15
20
25
30
5
10
15
20
25
0
0.2
0.4
0.6
0.8
Length (mm)
Spacing (mm)
Square EMCC M5 (%)
Figure 6 – Average and modal square EMCC for the sandwich
beam for varying patches spacing and length.
timal spacing depends on the eigenmode as expected. Hence,
the patch length is fixed at a = 5 mm but the spacing is left to
vary together with the patches thickness in a following sec-
ond analysis.
The patches thickness is then set to vary in the range
[0.25–1.00] mm. This means that the Aluminum/PZT thick-
ness ratio (H/h) may now be increased (varying in the range
[3–12]). This also means that the rigid foam thickness varies
together with the patches thickness. As in the previous anal-
ysis, the effective EMCC for each configuration is evaluated
using equation (66), for which all patches are set to either
short-circuit or open-circuit condition at the same time. Fig-
ure 7 presents the average and modal square EMCC for the
first five bending modes. It indicates that thicker piezoelec-
tric patches lead to higher EMCC. The reason for that might
be higher shear strains induced in the piezoelectric patches
and, consequently, higher induced electric potentials. As ob-
served in the previous analysis, the optimal spacing depends
on the eigenmode under consideration. However, it seems
that, for all cases, wider spacings between patches provide
better results (e = 20 − 30 mm).
In addition to the parametric analysis, an optimization
was also performed to include other design parameters, such
as the rigid foam stiffness, and to allow independent posi-
0.2
0.4
0.6
0.8
10
20
30
0
0.2
0.4
0.6
0.8
Thickness (mm)
Spacing (mm)
Average Square EMCC (%)
0.2
0.4
0.6
0.8
10
20
30
0
0.2
0.4
0.6
0.8
Thickness (mm)
Spacing (mm)
Square EMCC M1 (%)
0.2
0.4
0.6
0.8
10
20
30
0
0.2
0.4
0.6
0.8
Thickness (mm)
Spacing (mm)
Square EMCC M2 (%)
0.2
0.4
0.6
0.8
10
20
30
0
0.2
0.4
0.6
0.8
Thickness (mm)
Spacing (mm)
Square EMCC M3 (%)
0.2
0.4
0.6
0.8
10
20
30
0
0.2
0.4
0.6
0.8
Thickness (mm)
Spacing (mm)
Square EMCC M4 (%)
0.2
0.4
0.6
0.8
10
20
30
0
0.2
0.4
0.6
0.8
Thickness (mm)
Spacing (mm)
Square EMCC M5 (%)
Figure 7 – Average and modal square EMCC for the sandwich
beam for varying patches spacing and thickness.
Figure 8 – Optimal design to maximize the average effective
square EMCC for the sandwich beam.
tioning of each piezoelectric patch. Hence, the parameters
d and e appearing in Figure 5 are now relaxed so that the
spacing between patches and between the first patch and the
clamped end are considered as design parameters. Also, a
larger number of piezoelectric patches is allowed, so that it
should be possible to better fill the sandwich beam core with
active materials. In addition to the spacing between patches,
the rigid foam shear modulus and piezoelectric patches thick-
ness and length are considered as design parameters. For the
sake of simplicity, however, the length a is considered equal
for all piezoelectric patches. Other geometric properties, ac-
cording to Figure 5, are fixed at L = 220 mm and H = 3 mm.
Material properties of the two aluminum faces are fixed at
Young modulus 69 GPa and mass density 2690 kg m
−3
.
The optimization was performed using a genetic algo-
rithm search method, with a population size of 180 individ-
uals evolving for 40 generations. During each generation,
25 individuals suffer mutations in all of its properties and
7 arithmetic crossovers between two randomly selected in-
dividuals are performed. The performance index is defined
as the average square EMCC for the first six bending eigen-
modes. Figure 8 presents the optimal design obtained at the
end of 40 generations. Its parameters are 0.4 MPa for the
foam shear modulus, 1 mm thickness and 5 mm length for
the piezoelectric patches, and the following positions for the
patches (from patch left end to clamped end): [11, 38, 59, 84,
107, 129, 155, 174, 201] mm. The effective square EMCC
obtained for the first six bending modes are [0.07, 0.42, 0.81,
1.15, 1.39, 1.60] %, averaging 0.91%. The EMCC results
indicate that the shear piezoceramic patches provide a better
coupling for the 4th, 5th and 6th modes.
4. APPLICATIONS FOR ACTIVE AND PASSIVE VI-
BRATION CONTROL
In this section, some recent results on active, passive and
active-passive vibration control using piezoelectric materials
are presented.
4.1. Active vibration control
First, some results of the experimental implementation
of a simple control strategy, so-called DVF (Direct Velocity
Feedback), to evaluate the voltage to be applied to piezoelec-
tric actuators, using both extension and shear mechanisms,
are presented. For that, two active damped cantilever beams
were considered and are shown in Figure 9. They were con-
structed using aluminum sheets with length 280 mm, width
25 mm and thickness 3 mm.
The actively damped beam (Figure 9a), named Extension
Active (EA), was obtained by bonding a transversely poled,
or extension, PIC151 piezoceramic patch (PI Ceramic) on
one of the surfaces of the beam using an epoxy-based glue
(Araldite) cured at 60
o
C. The piezoceramic patch has length
25 mm, width 25 mm and thickness 0.5 mm and was bonded
at 70 mm of the aluminum sheet left end. The beam with
bonded piezoceramic patch is then clamped, using two bolted
thick steel plates fixed to a heavy concrete block, such that
the cantilever section is 220 mm long and the piezoceramic
patch is located at 10 mm of the clamp, as shown in Fig-
ure 9a. Another configuration, shown in Figure 9b and
named Shear Active-Passive (SAP), is a sandwich construc-
tion with two 3 mm aluminum facings. The sandwich core is
composed of a longitudinally poled, or shear, PIC255 piezo-
ceramic patch (PI Ceramic) and two layers of a viscoelastic
double coated polyethylene foam tape (3M 4494), as shown
in Figure 9d. The foam tape has thickness 1 mm and width
25 mm. The piezoceramic patch is 0.5 mm thick but this
thickness increases to approximately 0.8 mm after cabling
of the top and bottom electrodes. The shear piezoceramic
patch was bonded to the top and bottom aluminum facings
with Araldite cured at 60
o
C. To withstand the large clamping
force of the bolts, a 1 mm aluminum sheet was used for the
60 mm long clamped section core. The experimental designs
are depicted in Figure 10.
A general schematic representation and picture of the ex-
perimental setup are depicted in Figure 11. An accelerom-
eter (Brüel&Kjær 4375) was considered for both measure-
ment and observation of the structural response and feedback
control. Its placement was optimized to provide a satisfac-
tory reading of the structural response and to improve the
active control performance of each configuration (Figure 9).
The accelerometer output is passed through a signal condi-
tioner (Brüel&Kjær 2626) before acquisition. The structure
is both excited and controlled using the piezoceramic patch,
(a) Extension Active (EA)
160
25
220
3.0
0.5
AccelerometerPiezoceramic
Aluminum
10
(b) Shear Active-Passive (SAP)
130
25
220
3.0
1.0
Accelerometer
Piezoceramic
Aluminum
Aluminum
3.0
Foam
10
Figure 9 – Schematic representation of the cantilever beams
with EA and SAP damping treatments (dimensions in mm and
not in scale).
(a) Extension Active (EA) (b) Shear Active-Passive (SAP)
Figure 10 – Pictures of the cantilever beams with EA and SAP
damping treatments.
i.e. the active element of each configuration. For that, two
independent signals are sent to the piezoceramic patch as de-
scribed later on. Since piezoceramic actuators require rel-
atively high voltages to provide satisfactory actuation per-
formance, a power amplifier (Midé EL-1224) with a gain
20 V/V was used to connect the dSPACE output to the piezo-
ceramic patches. The dSPACE connector panel (CLP) input
and output connectors are limited to ±10 V. Hence, a max-
imum voltage amplitude of 200 V could be applied to the
piezoceramic patches. Since the piezoceramic patches are
0.5 mm thick, this leads to an electric field magnitude up
to 400 V/mm, which is sufficient to achieve significant ac-
tuation forces. Both accelerometer input and piezoceramic
output signals were processed using a DS1104 dSPACE con-
trol system connected to a PC. This allows to implement
both signal processing and control law using Simulink mod-
els which are then compiled through Real Time Workshop
and uploaded in the dSPACE controller board.
The Simulink model used for the signal processing and
control law implementation is shown in Figure 12. It consists
of four main groups of blocks: 1) acquisition, filtering and
integration of accelerometer signal to provide a clean trans-
verse velocity measurement at the accelerometer location; 2)
evaluation of the control voltage to be applied to the piezo-
ceramic patch using a simple direct velocity feedback law,
where the control gain can be chosen afterwards, followed by
a controlled saturation to avoid uncontrolled saturation of the
combined voltage signals at the dSPACE connector panel;
3) setup of the excitation voltage signal to be applied to the
piezoceramic patch, for which a chirp signal was used; and
4) output of combined voltage signals, to be applied to the
piezoceramic patch, to the dSPACE connector panel. Some
issues regarding the Simulink model deserve discussion. In
particular, the high-pass filtering of the accelerometer signal
ADC DAC
PA SC
dSPACE CLP
dSPACE Control Desk
SIMULINK/MATLAB
PZT
Accelerometer
Figure 11 – Schematic representation and picture of the experi-
mental setup.
was used mainly to eliminate the DC component and thus a
simple second-order Butterworth high-pass filter with a cut-
off frequency of 50 rad/s was considered. In order to allow
independent excitation and control through the piezoceramic
patch and avoid uncontrolled saturation, the control voltage
signal saturation was set up to 150 V while the chirp exci-
tation voltage, limited to 45 V, was defined by manually ad-
justing the chirp gain so that the structural response measure-
ment was optimized for each configuration. A sampling time
of 50 µs was considered for the acquisition.
Velocity Total
voltage
Saturation
−K−
SC gain
RTI Data
1/20
PA gain
0.1
Output gain
1
s
Integrator
10
Input gain
s
2
den(s)
High−pass lter
(50 rad/s)
Delay Chirp
DAC
DS1104DAC_C1
ADC
DS1104ADC_C6
−K−
Control gain
Control
voltage
20
Chirp gain
Chirp
voltage
Acceleration
1
2
3
4
Figure 12 – Simulink model used for signal processing and feed-
back control voltage evaluation.
Using the compiled model uploaded in the dSPACE con-
trolled board, a series of measurements were performed for
each design configuration to determine satisfactory parame-
ters of the data acquisition, signal processing, accelerometer
location and piezoceramic patches control gains. To qual-
ify the measurements and the damping performance, the fre-
quency response function (FRF) between the chirp excita-
tion voltage and the transverse velocity at the accelerometer
location was evaluated. For that, a routine was developed in
MATLAB to process data recorded by the dSPACE interface.
It consists mainly of evaluating and plotting the FRF for each
run, recording N selected satisfactory FRFs, and evaluating,
plotting and recording the average FRF. In this work, N = 20
selected FRFs were used to evaluate the average FRF. A FRF
estimator equal to the cross-spectrum, between input (chirp
voltage) and output (velocity), divided by the autospectrum
of the input was considered. The input and output spectra
were obtained using the fast fourier transform algorithm of
MATLAB.
10
2
10
3
−150
−140
−130
−120
−110
−100
−90
−80
−70
−60
−50
−40
Velocity/Voltage (m/s/V, dB)
Frequency (Hz)
k0
k1
k2
k3
Figure 13 – FRF for the EA damped cantilever beam with ac-
celerometer at 160 mm from clamped end for various control
gains: k
0
= 0, k
1
= −2 kVs/m, k
2
= −6 kVs/m, k
3
= −10 kVs/m.
10
2
10
3
−150
−140
−130
−120
−110
−100
−90
−80
−70
Velocity/Voltage (m/s/V, dB)
Frequency (Hz)
k0
k1
k2
k3
Figure 14 – FRF for the SAP damped sandwich beam with
accelerometer at 130 mm from clamped end for various con-
trol gains: k
0
= 0, k
1
= −10 kVs/m, k
2
= −25 kVs/m, k
3
=
−40 kVs/m.
Some damping performance analyses based on the FRFs
obtained for each configuration design and for four selected
control gains are now presented. In particular, Figures 13
and 14 show the FRF for the two cantilever beams: EA and
SAP, respectively. Figure 13 shows that the velocity feed-
back control law may yield a significant decrease in the re-
sponse amplitude at the first and second eigenfrequencies.
However, the same performance is not observed at the other
eigenfrequencies. In fact, the amplitude at the fifth eigenfre-
quency increases with the control gain magnitude. This was
expected and it is a known downside of such a simple control
law. It may be easier however to analyze the damping per-
formance of this active damping treatment through the modal
damping factors for each control gain. They are shown in Ta-
ble 3 for the first five bending modes of the EA damped can-
tilever beam. Notice that the beam without active damping
is represented by k
0
= 0, that is with zero control voltage.
In this case, the piezoceramic patch acts only as additional
mass and stiffness to the beam and, thus, should not improve
modal damping. Hence, as expected, very low damping fac-
tors (less than 0.4%) were measured for the zero control gain
and are attributed mainly to the material damping of the alu-
minum beam. This case may therefore be used as a refer-
ence for the non-treated cantilever beam. As observed in
Figure 13, the first and second modes damping factors are
significantly increased by the feedback controller, while the
fifth mode damping factor has a small overall decrease for the
larger control gain. Notice that, as it is well-known for direct
velocity feedback controllers, this damping performance is
valid only for this selected position of the accelerometer, at
160 mm from the clamp, for which the controller clearly pri-
oritizes the first vibration mode.
Table 2 – Damping factors (%) for the first five bending modes
of EA and SAP damped beams with various control gains
(kVs/m).
Gain M 1 M 2 M 3 M 4 M 5
k
0
= 0 0.39 0.17 0.16 0.25 0.25
EA k
1
= −2 4.12 1.29 0.21 0.63 0.32
k
2
= −6 10.9 3.83 0.27 1.20 0.27
k
3
= −10 14.9 6.99 0.33 1.71 0.22
k
0
= 0 3.49 1.85 0.90 0.98 1.25
SAP k
1
= −10 5.75 5.22 1.84 1.39 1.18
k
2
= −25 8.17 11.3 4.09 1.90 1.13
k
3
= −40 10.8 18.7 9.91 2.41 1.02
Then, the sandwich shear active-passive (SAP) damped
beam is analyzed. From its FRF, shown in Figure 14, it is
possible to observe that a significantly higher decrease in
resonant amplitudes can be obtained with this design. No-
tice however that care should be taken when comparing this
design with the other active-passive ones since, in this case,
the base structure is quite different. Still, Figure 14 shows
that a significant decrease in the response amplitude can be
obtained at the first four eigenfrequencies. This fact is con-
firmed by the modal damping factors, presented in Table 3,
where it can be observed that, although the third and fourth
modes are less damped passively, large damping factors can
be obtained by increasing the control gain. It is also notice-
able that the SAP treatment does not affect much the fifth
vibration mode.
4.2. Passive vibration control
Passive vibration control can be obtained using a variety
of shunt electric circuits connected to the piezoelectric ele-
ments. The general idea is to dissipate part of the vibratory
energy through conversion to electric energy by the piezo-
electric elements and then dissipation in the circuit compo-
nents, such as dissipation via Joule effect in a resistor. Two
simple shunt circuits, largely studied in the literature, are the
resistive and resistive-inductive (also called resonant) ones.
A simple analysis of the effect these components may have
in the coupled piezoelectric structure can be done using the
electric charge formulation presented previously.
Let us suppose a harmonic mechanical excitation, such
that, in (41),
V
c
= 0 ; F
m
= b
˜
f e
jωt
; u =
˜
ue
jωt
; q
p
=
˜
q
p
e
jωt
(68)
such that the equations of motion, defining M = M
s
+ M
p
,
can be rewritten as
(−ω
2
M + K
us
+ K
D
up
)
˜
u − K
uq
˜
q
p
= b
˜
f
−K
t
uq
˜
u + (−ω
2
L
c
+ jωR
c
+ K
q
)
˜
q
p
= 0
(69)
Solving the second equation of (69) for
˜
q
p
and substitut-
ing in the first equation yields
−ω
2
M + K
us
+ K
D
up
−K
uq
(−ω
2
L
c
+ jωR
c
+ K
q
)
−1
K
t
uq
˜
u = b
˜
f (70)
Supposing the measurement of a selected displacement in
the structure, defined as
˜y = c
˜
u (71)
where c is an output distribution vector, the complex fre-
quency response function of the displacement output when
subjected to the mechanical force input can be defined such
as
˜y = G(ω)
˜
f (72)
where
G(ω) = c
−ω
2
M + K
us
+ K
D
up
−K
uq
(−ω
2
L
c
+ jωR
c
+ K
q
)
−1
K
t
uq
−1
b (73)
Notice that the resistance and inductance of the electric
circuits lead to a modification of the dynamic stiffness of
the piezoelectric elements. The short-circuited and open-
circuited cases can be represented by considering L
c
= R
c
=
0 and R
c
→ ∞, respectively. In these cases, the frequency
response function reduces to
G(ω) = c
−ω
2
M + K
us
+ K
D
up
− K
uq
K
−1
q
K
t
uq
−1
b (74)
and
G(ω) = c
−ω
2
M + K
us
+ K
D
up
−1
b (75)
respectively. In the first case, there is a reduction of the
piezoelectric elements stiffnesses due to the relaxation of the
induced difference of electric potential, while, in the second
case, the piezoelectric elements stiffnesses are those for con-
stant electric displacements. However, in both cases, the re-
sulting stiffness is real and constant. On the other hand, if
resistive shunt circuits are considered (L
c
= 0, but R
c
6= 0),
the frequency response function is modified to
G(ω) = c
−ω
2
M + K
us
+ K
D
up
−K
uq
(jωR
c
+ K
q
)
−1
K
t
uq
−1
b (76)
this leads to complex-valued stiffness matrices for the piezo-
electric elements, hence, hysteretic damping is added to the
structure. Proper tuning of the resistance for a given shunt
circuit can modify the resonance frequency and damping of
a selected vibration mode. Clearly, the higher the electrome-
chanical coupling, through K
uq
, the higher the damping that
can be achieved.
When a shunt circuit with both resistance and inductance
is considered, (73) indicates that an additional resonance may
appear in the frequency response function. Indeed, the cou-
pling of a resonant shunt circuit (resistive-inductive) to a
piezoelectric structure may function as a dynamic vibration
absorber. Thus, by properly tuning the resistance and induc-
tance of a given shunt circuit, it is possible to absorb part of
the vibratory energy of a selected vibration mode and then
dissipate this energy through the resistance.
Modal damping using a resonant shunted piezoelectric
patch. To illustrate these passive vibration control ef-
fects, the frequency response of a cantilever beam with
a bonded extension piezoelectric patch connected to a
resistive-inductive shunt circuit (Figure 15) is analyzed. The
host beam, of width 25 mm, is made of aluminum with
Young modulus 70 GPa and mass density 2700 kg m
−3
. The
piezoelectric material is a PZT-5H with elastic coefficient
(at constant electric displacement) 97.8 GPa, mass density
7500 kg m
−3
, piezoelectric coefficient -1.35 GN C
−1
and di-
electric coefficient (at constant strain) 99.74 Mm F
−1
.
25
220
3.0
0.5
Piezoelétrico
Aluminio
10
Figure 15 – Cantilever beam with a bonded extension piezoelec-
tric patch connected to a resistive-inductive shunt circuit.
Figure 16a shows the frequency response function of
the cantilever beam, zoomed around the first resonance fre-
quency, for four circuit conditions: open-circuit (R
c
→ ∞,
solid), short-circuit (R
c
= L
c
= 0, long dash), resistive (R
c
=
9223 Ω, L
c
= 0, short dash) and resistive-inductive or reso-
nant (R
c
= 3452 Ω, L
c
= 691 H, dash-dot). The resistance
35 40 45 50 55 60 65 70 7
5
−10
−5
0
5
10
15
20
25
30
Velocidade na ponta (m/s/N, dB)
Frequência (Hz)
(a)
320 325 330 335 340 345 35
0
−5
0
5
10
15
Velocidade na ponta (m/s/N, dB)
Frequência (Hz)
(b)
Figure 16 – Frequency response function of the tip velocity of
a cantilever beam subjected to a mechanical force, zoomed at
first (a) and second (b) resonance frequencies, for four circuit
conditions: open-circuit (solid), short-circuit (long dash), resis-
tive (short dash) and resistive-inductive (dash-dot).
of the resistive circuit was tuned to maximize the shunted
piezoelectric loss factor at the first resonance frequency [38],
while the resistance and inductance of the resonant circuit
were tuned to position an anti-resonance at the first reso-
nance frequency and limit the amplitudes at the two adjacent
resonances, according to the standard design of dynamic ab-
sorbers [38]. One may notice that, as expected, short-circuit
condition makes the piezoelectric element less stiff and, thus,
reduces the structure resonance frequency compared to the
open-circuit condition. The purely resistive shunt circuit
leads to an intermediate resonance frequency, between short-
circuit and open-circuit ones, and adds some damping to the
structure, reducing the amplitude at resonance by approxi-
mately 5 dB. The resonant circuit however allows a much
greater reduction of the amplitude at resonance (more than
20 dB) without increasing much the amplitude outside the
resonance region. Notice also that unlike the impedance for-
mulation considered in [38], the present formulation allows
the analysis of effects at other frequency bands. Figure 16b
shows the frequency response function, zoomed around the
second resonance frequency, where one may observe that the
resistive circuit yields some amplitude reduction for the sec-
ond resonance also while the resonant circuit does not affect
the second resonance, which keeps its open-circuit condition.
Multimodal damping using several shunted piezoelectric
patches. Here, a cantilever sandwich beam with three shear
piezoelectric patches connected to three independent resis-
tive shunt circuits embedded in its foam core is considered.
The test structure is shown in Figure 17. The geometrical
properties are d = 20 mm and L = 25 mm. The material
properties are Young’s modulus 210 GPa, Poisson’s ratio 0.3
and density 7850 kg m
−3
for the steel; Young’s modulus
35.3 MPa, shear modulus 12.76 MPa and density 32 kg m
−3
for the rigid foam; Young’s modulus 61.1 GPa (SC), shear
modulus 23 GPa (SC), density 7500 kg m
−3
, piezoelectric
coefficient e
15
17 C m
−2
and constant stress dielectric coef-
ficient
T
11
27.7 nF m
−1
for the PZT-5H piezoceramic mate-
rial. Based on these PZT-5H material properties, a maximum
achievable loss factor with properly tuned resistive shunt cir-
cuit is around 31%. For comparison purposes, the uncon-
trolled beam is supposed to have a constant modal damping
factor of 0.16%, representing damping sources other than the
shunt circuits.
100 mm
d L e L e L
5.0 mm
5.0 mm
0.5 mm
Steel
Steel
PZT-5H
Rigid Foam
PZT-5H PZT-5H
Figure 17 – Schematic representation of the cantilever sandwich
beam with three piezoelectric patches (not in scale).
A parametric analysis was performed to evaluate the ef-
fects of the distance d, spacing e and length L of the piezo-
electric patches on the passive shunted damping. For each
of the lengths, the distance from the clamp and the spac-
ing between patches were varied defining a set of geomet-
ric configurations. For each configuration, the choice of the
eigenmodes to be damped by each piezoelectric patch was
performed by, first, tuning each shunted piezoelectric patch
to the three selected eigenmodes (3rd, 4th and 5th), one at
a time. Then, the damping factors provided for each eigen-
mode and for each patch are compared and the pair patch-
eigenmode leading to the highest damping factor is selected.
This procedure is repeated until all piezoelectric patches are
assigned to one eigenmode each. This procedure aims to as-
sure patch-eigenmode assignments that maximize the over-
all damping for each geometric configuration. Then, using
the three selected patch-eigenmode tuned pairs, the modal
damping factors for the third, fourth and fifth eigenmodes are
evaluated. It was observed that smaller distances yield higher
average damping while the optimal spacing depends on the
patches length. Higher average damping factor is obtained
for L = 20 mm length patches, spaced by e = 14 mm and
Table 3 – Damping factors for the second to fifth eigenmodes with individual contributions from each patch and loss factors of each
patch at each eigenfrequency.
Damping factor (%) Loss factor (%)
Modes Frequency (Hz) Patch 1 Patch 2 Patch 3 Total Patch 1 Patch 2 Patch 3
2 5258 0.13 0.07 0.08 0.28 8.66 13.07 19.90
3 14399 0.31 0.39 0.24 0.94 20.96 27.40 30.86
4 23604 0.65 0.77 0.45 1.87 28.11 30.86 27.40
5 36318 0.93 0.44 0.43 1.80 30.86 28.11 20.96
with the first patch d = 3 mm distant from the clamp. How-
ever, average damping factors around 1.5% can be observed
throughout the ranges d = 3 − 7 mm and e = 11 − 14 mm.
For the optimal configuration, the average damping factor is
1.54%, while the individual modal damping factors for the
third, fourth and fifth modes are, respectively, 0.94%, 1.87%
and 1.80%. This damping performance was obtained by tun-
ing the piezoelectric patches P1, P2 and P3 to the 5th, 4th
and 3rd eigenmodes, respectively.
Table 3 shows the breakdown of the modal damping fac-
tors with individual contributions from each piezoelectric
patch to each eigenmode. It can be observed that although
each patch was tuned to only one eigenmode, all patches
contribute to all eigenmodes, including the 2nd eigenmode
which was not included in the tuning. As expected, the
largest contributions for the fifth and fourth eigenmodes
damping were produced by their assigned patches, P1 and
P2, respectively. However, the same behavior was not ob-
served for the third eigenmode, for which the smaller con-
tribution comes from its assigned patch P3. This can be ex-
plained by the fact that the third eigenmode was the last one
to have a patch assigned to it, due to its overall smaller damp-
ing, and thus it was assigned to the last patch available (P3).
10
3
10
4
10
5
0
5
10
15
20
25
30
35
Frequency (Hz)
Damping factor (%)
ω
3
→
ω
4
↓
← ω
5
R
op
3
= 436.06Ω
R
op
4
= 265.09Ω
R
op
5
= 170.84Ω
Figure 18 – Loss factor of the three shunted piezoelectric
patches tuned to the 3rd, 4th and 5th eigenmodes as a function
of frequency.
It is worth noticing that the significant cross-contribution
between patches is obtained thanks to the wide frequency
range of the shunted patches loss factors, due to the small
optimal electric resistance of the corresponding shunt cir-
cuits. This fact can also be observed from Figure 18 and
Table 3. In particular, Table 3 presents also the loss factor
of each shunted piezoelectric patch when excited at the sec-
ond to fifth eigenfrequencies and Figure 18 shows these loss
factors as functions of frequency. It can be observed that, as
expected, the maximum loss factor (30.86%) for each patch
is obtained at its corresponding tuned eigenfrequency. How-
ever, significant loss factor values are maintained at the other
eigenfrequencies. In particular, a minimum loss factor of al-
most 21% is obtained at the 3rd, 4th and 5th eigenfrequencies
for all patches. Even at the second eigenfrequency, which
was not included in the tuning procedure, loss factors up to
20% are obtained for the third patch.
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10
4
−120
−100
−80
−60
−40
−20
0
20
Frequency (Hz)
Tip velocity (m/s/N, dB)
R
sc
R
op
R
oc
Figure 19 – Frequency response function of beam tip veloc-
ity with short-circuited (sc), open-circuited (oc) and optimally
shunted (op) piezoelectric patches.
The frequency response function of the sandwich beam
tip velocity, when excited by a transversal force applied at
the same point, was evaluated and is shown in Figure 19 for
the following electric boundary conditions: SC – all patches
in short-circuit, OC – all patches in open-circuit and OP – all
patches optimally shunted.
From Figure 19, it is possible to observe that a reduction
of approximately 20 dB can be achieved in the amplitude at
resonance for the 3rd, 4th and 5th eigenmodes, which were
prioritized by the shunted piezoelectric patches. In addition,
a reduction of approximately 10 dB is observed for the 2nd
eigenmode. These results indicate that not only wider fre-
quency ranges are achievable but also the damping perfor-
mance of each eigenmode is increased when using a larger
number of piezoelectric patches.
4.3. Active-passive vibration control
As an extension of the use of shunt circuits connected
to piezoelectric elements, it is possible to include a volt-
age source to these circuits so that the piezoelectric element
may act both as vibration dampers/absorbers and actuators.
Hence, passive, active and active-passive vibration control
can be obtained. In the particular case of the circuit pro-
posed previously, composed of a resistance, an inductance
and a voltage source, it is worthwhile to analyze the active
and active-passive action, since the passive action is obtained
by eliminating the voltage source, which is the case treated in
the previous section. Therefore, a harmonic excitation anal-
ysis similar to the one presented in the previous section can
be done here, but considering an excitation through a single
voltage source, such that
F
m
= 0 ; V
c
=
˜
V
c
e
jωt
; u =
˜
ue
jωt
; q
p
= ˜q
p
e
jωt
(77)
The equations of motion (41) can be rewritten as
(−ω
2
M + K
us
+ K
D
up
)
˜
u − K
uq
˜q
p
= 0
−K
t
uq
˜
u + (−ω
2
L
c
+ jωR
c
+ K
q
) ˜q
p
=
˜
V
c
(78)
Solving the second equation of (78) for ˜q
p
and replacing
the solution in the first equation leads to
−ω
2
M + K
us
+ K
D
up
−K
uq
(−ω
2
L
c
+ jωR
c
+ K
q
)
−1
K
t
uq
˜
u
= K
uq
(−ω
2
L
c
+ jωR
c
+ K
q
)
−1
˜
V
c
(79)
As in the previous section, supposing the measurement of
a selected displacement in the structure, defined as
˜y = c
˜
u (80)
the complex frequency response function of the displacement
output when subjected to the voltage input can be defined
such as
˜y = G(ω)
˜
V
c
(81)
where
G(ω) = c
−ω
2
M + K
us
+ K
D
up
−K
uq
(−ω
2
L
c
+ jωR
c
+ K
q
)
−1
K
t
uq
−1
× K
uq
(−ω
2
L
c
+ jωR
c
+ K
q
)
−1
(82)
As in the passive case studied previously, the resistance
and inductance of the electric circuits lead to a modification
of the dynamic stiffness of the piezoelectric elements. The
purely active case can be represented by considering L
c
=
R
c
= 0. In this case, the frequency response function reduces
to
G(ω) = c
−ω
2
M + K
us
+ K
D
up
− K
uq
K
−1
q
K
t
uq
−1
K
uq
K
−1
q
(83)
where K
uq
K
−1
q
indicates the equivalent force vector induced
per voltage applied to the actuator and the reduction of piezo-
electric stiffness accounts for the modification from a con-
stant electric displacement to a constant electric field condi-
tion. The open-circuit case (R
c
→ ∞) leads to the impossibil-
ity of actuation as expected.
For the most general case, the inductance and resistance
not only modify the dynamic stiffness of the piezoelectric
element, leading to damping and/or absorption, but also af-
fects the active control authority of the actuator due to the
term K
uq
(−ω
2
L
c
+ jωR
c
+ K
q
)
−1
in (82). In particular, it
may be possible to amplify the active control authority near
the resonance frequency of the electric circuit [46, 53, 54].
To illustrate these active and active-passive vibration con-
trol effects, the same cantilever beam studied previously is
considered (Figure 15). However, a voltage source is in-
cluded in series with the inductance and resistance. The con-
trol voltage is evaluated using a Linear Quadratic Regulator
algorithm (state feedback), for which a unitary state weight
matrix Q = I is considered and the control input weight is
adjusted so that the control voltage is limited to 200 V.
First, an analysis of the active control authority is per-
formed. Figure 20 shows the frequency response function of
the tip velocity of the cantilever beam subjected to a voltage
applied to the circuit. It shows that the resistive shunt cir-
cuit diminishes the active control authority for all frequen-
cies. This is quite reasonable since part of the input electric
energy is being redirected and dissipated through the resis-
tance. On the other hand, the resonant shunt circuit allows
an increase of the active control authority around the first
resonance, although it decreases the authority exactly at res-
onance frequency (Figure 20b), at the cost of reducing the ac-
tive control authority for the remaining frequency band (Fig-
ure 20a).
Figure 21a shows the frequency response function of
the cantilever beam, zoomed around the first resonance
frequency, for the uncontrolled beam (open-circuit, R
c
→
∞, solid), passive controlled beam with resistive (R
c
=
9223 Ω, L
c
= 0, V
c
= 0, short dash) and resonant (R
c
=
3452 Ω, L
c
= 691 H, V
c
= 0, short dash-dot) shunt cir-
cuits, active controlled beam (R
c
= L
c
= 0, V
c
< 200 V ,
long dash) and active-passive controlled beam with resistive
(R
c
= 9223 Ω, L
c
= 0, V
c
< 200 V , long dash-dot) and res-
onant (R
c
= 3452 Ω, L
c
= 691 H, V
c
< 200 V , dash-circle)
shunt circuits. The purely active control adds some (active)
damping to the structure reducing the amplitude at resonance
by approximately 8 dB. The active-passive control yields bet-
ter performances with amplitude reduction of approximately
11 dB, for the resistive, and 25 dB, for the resonant. No-
tice that despite the small reduction on the active control
authority exactly at resonance frequency, the active-passive
control always outperform the corresponding passive (open-
loop) condition, that is the active controller reduces ampli-
tude further from the corresponding passive circuit. Figure
21b shows the frequency response function, zoomed around
10
2
10
3
−160
−140
−120
−100
−80
−60
−40
Velocidade na ponta (m/s/V, dB)
Frequência (Hz)
(a)
20 30 40 50 60 70 80 90 100
−90
−85
−80
−75
−70
−65
−60
−55
−50
−45
−40
−35
Velocidade na ponta (m/s/V, dB)
Frequência (Hz)
(b)
Figure 20 – Comparison of active control authority for the
purely active and resistive and inductive active-passive circuits
for whole frequency range (a) and zoomed around the first res-
onance frequency.
the second resonance frequency, where one may observe that
the purely active and resistive, passive, active and active-
passive, circuits yield some amplitude reduction for the sec-
ond resonance also while the resonant circuit does not allow
control of the second resonance, since it limits both active
and passive effects to a narrow frequency-band around the
first resonance, for which the inductance was tuned.
5. CONCLUDING REMARKS
The present article has presented a brief review of the
open literature concerning applications of piezoelectric sen-
sors and actuators for active and passive vibration control.
Some recent advances on this subject were also presented
with special attention to the following aspects: i) modeling of
structures with piezoelectric sensors and actuators, ii) evalu-
ation of the effective electromechanical coupling coefficient,
iii) applications for active and passive vibration control. It
was shown that piezoelectric materials can be quite effec-
tive as distributed sensors and actuators for active, passive
35 40 45 50 55 60 65 70 7
5
−10
−5
0
5
10
15
20
25
30
Velocidade na ponta (m/s/N, dB)
Frequência (Hz)
(a)
320 325 330 335 340 345 35
0
−5
0
5
10
15
Velocidade na ponta (m/s/N, dB)
Frequência (Hz)
(b)
Figure 21 – Frequency response function of the tip velocity
of a cantilever beam subjected to a mechanical force, zoomed
at first (a) and second (b) resonance frequencies, for: open-
circuit (solid), passive resistive (short dash), passive resistive-
inductive (short dash-dot), active (long dash), active-passive
resistive (long dash-dot) and active-passive resistive-inductive
(dash-circle).
and active-passive vibration control. In particular, impor-
tant design parameters include: effective electromechanical
coupling coefficient, actuation/sensing mechanism or mode,
patches size and location. It was also shown that proper de-
sign should include electromechanical coupled models for
the ensemble structure-patches-circuits.
ACKNOWLEDGEMENTS
This research was supported by the State of São Paulo
Research Foundation (FAPESP), through research grant
04/10255-7, which is gratefully acknowledged. The author
also thanks Prof. Ayech Benjeddou for his valuable inputs
and the authors’ students Heinsten Santos, Rafael Azevedo,
Carlos Maio for generating part of the presented results.
REFERENCES
[1] M. Sunar and S.S. Rao. Recent advances in sensing and
control of flexible structures via piezoelectric materi-
als technology. Applied Mechanics Review, 52(1):1–16,
1999.
[2] M. Ahmadian and A.P. DeGuilio. Recent advances in
the use of piezoceramics for vibration suppression. The
Shock and Vibration Digest, 33(1):15–22, 2001.
[3] S.O. Reza Moheimani. A survey of recent innovations
in vibration damping and control using shunted piezo-
electric transducers. IEEE Transactions on Control Sys-
tems Technology, 11(4):482–494, 2003.
[4] I. Chopra. Review of state of art of smart structures and
integrated systems. AIAA Journal, 40(11):2145–2187,
2002.
[5] M.A. Trindade and A. Benjeddou. Hybrid active-
passive damping treatments using viscoelastic and
piezoelectric materials: review and assessment. Jour-
nal of Vibration and Control, 8(6):699–746, 2002.
[6] C.T. Sun and X.D. Zhang. Use of thickness-shear mode
in adaptive sandwich structures. Smart Materials and
Structures, 4(3):202–206, 1995.
[7] A. Benjeddou, M.A. Trindade, and R. Ohayon. A uni-
fied beam finite element model for extension and shear
piezoelectric actuation mechanisms. Journal of Intel-
ligent Material Systems and Structures, 8(12):1012–
1025, 1997.
[8] A. Benjeddou, M.A. Trindade, and R. Ohayon. New
shear actuated smart structure beam finite element.
AIAA Journal, 37(3):378–383, 1999.
[9] A. Benjeddou, M.A. Trindade, and R. Ohayon. Piezo-
electric actuation mechanisms for intelligent sandwich
structures. Smart Materials and Structures, 9(3):328–
335, 2000.
[10] O.J. Aldraihem and A.A. Khdeir. Smart beams with
extension and thickness-shear piezoelectric actuators.
Smart Materials and Structures, 9(1):1–9, 2000.
[11] A.A. Khdeir and O.J. Aldraihem. Deflection analysis of
beams with extension and shear piezoelectric patches
using discontinuity functions. Smart Materials and
Structures, 10(2):212–220, 2001.
[12] O.J. Aldraihem and A.A. Khdeir. Exact deflection solu-
tions of beams with shear piezoelectric actuators. Inter-
national Journal of Solids and Structures, 40(1):1–12,
2003.
[13] M.A. Trindade, A. Benjeddou, and R. Ohayon. Para-
metric analysis of the vibration control of sandwich
beams through shear-based piezoelectric actuation.
Journal of Intelligent Material Systems and Structures,
10(5):377–385, 1999.
[14] S. Raja, R. Sreedeep, and G. Prathap. Bending behav-
ior of hybrid-actuated piezoelectric sandwich beams.
Journal of Intelligent Material Systems and Structures,
15(8):611–619, 2004.
[15] B.P. Baillargeon and S.S. Vel. Active vibration suppres-
sion of sandwich beams using piezoelectric shear actu-
ators: experiments and numerical simulations. Jour-
nal of Intelligent Material Systems and Structures,
16(6):517–530, 2005.
[16] S.S. Vel and R.C. Batra. Exact solution for the cylindri-
cal bending of laminated plates with embedded piezo-
electric shear actuators. Smart Materials and Struc-
tures, 10(2):240–251, 2001.
[17] L. Edery-Azulay and H. Abramovich. Piezoelectric
actuation and sensing mechanisms - closed form solu-
tions. Composite Structures, 64(3-4):443–453, 2004.
[18] M.A. Trindade. Simultaneous extension and shear
piezoelectric actuation for active vibration control of
sandwich beams. Journal of Intelligent Material Sys-
tems and Structures, 18(6):591–600, 2007.
[19] A. Benjeddou. Shear-mode piezoceramic advanced ma-
terials and structures: a state of the art. Mechanics
of Advanced Materials and Structures, 14(4):263–275,
2007.
[20] G.A. Lesieutre and C.L. Davis. Can a coupling coeffi-
cient of a piezoelectric device be higher than those of
its active material? Journal of Intelligent Material Sys-
tems and Structures, 8(10):859–867, 1997.
[21] W.P. Mason. Piezoelectric crystals and their applica-
tion to ultrasonics. Van Nostrand, New York, 1950.
[22] A.F. Ulitko. On the theory of electromechanical con-
version of energy in nonuniformly deformed piezoce-
ramic bodies. Prikladnaya Mekhanika, 13(10):115–
123, 1977.
[23] S.H. Chang, N.N. Rogacheva, and C.C. Chou. Analysis
of methods for determining electromechanical coupling
coefficients of piezoelectric elements. IEEE Trans-
actions on Ultrasonics, Ferroelectrics, and Frequency
Control, 42(4):630–640, 1995.
[24] B. Aronov. On the optimization of the effective elec-
tromechanical coupling coefficients of a piezoelectric
body. Journal of the Acoustical Society of America,
114(2):792–800, 2003.
[25] N.N. Rogacheva, C.C. Chou, and S.H. Chang. Elec-
tromechanical analysis of a symmetric piezoelec-
tric/elastic laminate structure: theory and experiment.
IEEE Transactions on Ultrasonics, Ferroelectrics, and
Frequency Control, 45(2):285–294, 1998.
[26] S. Basak, A. Raman, and S.V. Garimella. Dynamic
response optimization of piezoelectrically excited thin
resonant beams. Journal of Vibration and Acoustics,
127(1):18–27, 2005.
[27] H. Boudaoud, A.Benjeddou, E.M. Daya, and S. Belou-
ettar. Analytical evaluation of the effective emcc of
sandwich beams with a shear-mode piezoceramic core.
In Proceedings of Second International Conference on
Desing and Modelling of Mechanical Systems, Monas-
tir, Tunisia, March 19-21 2007.
[28] M. Naillon, R.H. Coursant, and F. Besnier. Analyse de
structures piézoélectriques par une méthode d’éléments
finis. Acta Electronica, 25(4):341–362, 1983.
[29] T. Bailey and J.E. Hubbard Jr. Distributed
piezoelectric-polymer active vibration control of a can-
tilever beam. Journal of Guidance, 8(5):605–, 1985.
[30] E.F. Crawley and J. deLuis. Use of piezoelectric actua-
tors as elements of intelligent structures. AIAA Journal,
25(10):1373–1385, 1987.
[31] E.F. Crawley and E.H. Anderson. Detailed models of
piezoceramic actuation of beams. Journal of Intelligent
Material Systems and Structures, 1(1):4–25, 1990.
[32] M.-H. Herman Shen. Analysis of beams containing
piezoelectric sensors and actuators. Smart Materials
and Structures, 3(4):439–447, 1994.
[33] C.K. Lee. Theory of laminated piezoelectric plates for
the design of distributed sensors/actuators. part i: gov-
erning equations and reciprocal relationships. Journal
of the Acoustical Society of America, 87:1145
˝
U1157,
1990.
[34] D.A. Saravanos and P.R. Heyliger. Mechanics and com-
putational models for laminated piezoelectric beams,
plates, and shells. Applied Mechanics Review,
52(10):305–320, 1999.
[35] S.V. Gopinathan, V.V. Varadan, and V.K. Varadan. A
review and critique of theories for piezoelectric lami-
nates. Smart Materials and Structures, 9:24–48, 2000.
[36] A. Benjeddou. Advances in piezoelectric finite el-
ements modeling of adaptive structural elements: a
survey. Computers and Structures, 76(1-3):347–363,
2000.
[37] R.L. Forward. Electronic damping of vibrations in op-
tical structures. Applied Optics, 18(5):690–697, 1979.
[38] N.W. Hagood and A. von Flotow. Damping of struc-
tural vibrations with piezoelectric materials and passive
electrical networks. Journal of Sound and Vibration,
146(2):243–268, 1991.
[39] C.H. Park and D.J. Inman. Enhanced piezoelectric
shunt design. Shock and Vibration, 10:127–133, 2003.
[40] F.A.C. Viana and V. Steffen Jr. Multimodal vibration
damping through piezoelectric patches and optimal res-
onant shunt circuits. Journal of the Brazilian Society of
Mechanical Sciences, 28(3):293–310, 2006.
[41] S. Raja, G. Prathap, and P.K. Sinha. Active vibration
control of composite sandwich beams with piezoelec-
tric extension-bending and shear actuators. Smart Ma-
terials and Structures, 11(1):63–71, 2002.
[42] A. Benjeddou and J.-A. Ranger-Vieillard. Passive vi-
bration damping using shunted shear-mode piezoce-
ramics. In B.H.V. Topping and C.A. Mota Soares, ed-
itors, Proceedings of the Seventh International Confer-
ence on Computational Structures Technology, page #4,
Stirling, Scotland, 2004. Civil-Comp Press.
[43] M.A. Trindade and C.E.B. Maio. Passive vibration con-
trol of sandwich beams using shunted shear piezoelec-
tric actuators. In IV ABCM National Congress of Me-
chanical Engineering, Recife, Brazil, August 2006.
[44] M.A. Trindade and C.E.B. Maio. Multimodal passive
vibration control of sandwich beams with shunted shear
piezoelectric materials. In 19th International Congress
of Mechanical Engineering (COBEM), Brasília, 2007.
[45] J. Tang, Y. Liu, and K.W. Wang. Semiactive and active-
passive hybrid structural damping treatments via piezo-
electric materials. The Shock and Vibration Digest,
32(3):189–200, 2000.
[46] M.S. Tsai and K.W. Wang. On the structural damp-
ing characteristics of active piezoelectric actuators
with passive shunt. Journal of Sound and Vibration,
221(1):1–22, 1999.
[47] A. Benjeddou and J.-A. Ranger. Use of shunted shear-
mode piezoceramics for structural vibration passive
damping. Computers and Structures, 84:1415–1425,
2006.
[48] The Institute of Electrical and Electronics Engineers
Inc. IEEE Standard on Piezoelectricity n.176-1987.
1987.
[49] The Institute of Radio Engineers. IRE standards on
piezoelectric crystals: Determination of the elastic,
piezoelectric, and dielectric constants - the electrome-
chanical coupling factor. Proceedings of the IRE,
46(4):764–778, 1958.
[50] C.D. Johnson, D.A. Kienholz, and L.C. Rogers. Fi-
nite element prediction of damping in beams with con-
strained viscoelastic layers. Shock and Vibration Bul-
letin, 50(1):71–81, 1981.
[51] G. Chevallier, J.-A. Ranger, A.Benjeddou, and H. Seb-
bah. Passive vibration damping using resistively
shunted piezoceramics: an experimental performance
evaluation. In Proceedings of Second International
Conference on Desing and Modelling of Mechanical
Systems, Monastir, Tunisia, March 19-21 2007.
[52] W.G. Cady. Piezoelectricity: An introduction to the the-
ory and applications of electromechanical phenomena
in crystals. McGraw-Hill, New York, 1946.
[53] C. Niezrecki and H.H. Cudney. Improving the power
consumption characteristics of piezoelectric actuators.
Journal of Intelligent Material Systems and Structures,
5(4):522–529, 1994.
[54] J. Sirohi and I. Chopra. Actuator power reduction using
L-C oscillator circuits. Journal of Intelligent Material
Systems and Structures, 12(12):867–877, 2001.
