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Active control of beam structures with piezoelectric
actuators and sensors: modeling and simulation
Isabelle Bruant, G. Cognal, F. Léné, M. Vergé
To cite this version:
Isabelle Bruant, G. Cognal, F. Léné, M. Vergé. Active control of beam structures with piezoelectric
actuators and sensors: modeling and simulation. Smart Materials and Structures, IOP Publishing,
2001, 10 (2), pp.404-408. �10.1088/0964-1726/10/2/402�. �hal-01978408�
TECHNICAL NOTE
Active control of beam structures with
piezoelectric actuators and sensors:
modeling and simulation
I Bruant, G Coffignal, F L
´
en
´
e and M Verg
´
e
Laboratoire de Mod
´
elisation et M
´
ecanique des Structures, CNRS UPRES A 8007, 8 rue du
Capitaine Scott, 75015 Paris, France
Abstract
The
active damping of structures is an important emerging field. In this
context, it is necessary to be able to develop new control methods for
flexible structures and simulate their effects. In order to be able to deal with
the optimization of active device locations, spillover and any other general
problems linked to control and model reduction, a simple but sufficiently
rich model is very useful. This is the reason why this technical note deals
with the modeling and simulation of the active vibration control of beam
structures using piezoelectric actuators and sensors.
In order to model beam structures equipped with piezoelectric devices,
we develop a simple finite composite beam element, taking into account the
properties of piezoelectric elements. This model uses six mechanical
degrees of freedom and four electric degrees of freedom. Then, a linear
quadratic regulator method is used to compute the control, including the
implementation of a state observer. Several simulations are presented.
1. Introduction
This technical note deals with the modeling of beam structures
equipped with piezoelectric devices (actuators and sensors).
In the case of a simple beam equipped with one piezoelectric
actuator and one sensor an analytical study can easily be
developed; but, in cases of structures made with several
beams equipped with actuators and sensors, a discrete model
is necessary in order to take into account the piezoelectric
effect. Moreover, most of piezoelectric finite elements are
three dimensional or two dimensional [1–4]. The use of these
finite models in the case of beam structures is not optimal,
in particular to solve structural optimization problems which
are computationally expensive. Finally, the active damping of
structures is an important emerging field. In this context, it is
necessary to be able to develop new control methods for flexible
structures and simulate their effects. In order to be able to
deal with the optimization of active device locations, spillover
and any other general problems linked to control and model
reduction, a simple but sufficiently rich model is very useful.
For this purpose, a simple finite beam element is described.
It corresponds to a composite beam made up of three layers:
one elastic and two piezoelectric. In this study, to simplify
the presentation, we consider a 2D beam finite element, but all
developments can be generalized to a 3D beam finite element.
Six mechanical degrees of freedom and four electric degrees of
freedom are used. From variational principles, the generalized
discrete equations are obtained. In order to set up the active
control the second differential equations are transformed into
a state space model. Then, principles of control theory can
be used: a linear quadratic control method, including a state
observer, is considered. Several simulations are presented.
The first one shows active damping of a simple cantilever beam
using one piezoelectric actuator and one sensor. It gives a
first validation of the finite element beam. Others simulations
consider a flexible three-beam structure.
2. Finite element formulation
Consider a flexible elastic beam structure, as shown in figure 1,
controlled by several piezoelectric actuators and sensors. Each
device is made up of a pair of piezoelectric materials, polarized
in the thickness direction and attached symmetrically. The
top and bottom sides of each piezoelectric are covered by
Figure 1. A three-beam structure.
electrodes to ensure the connection with the electric circuit. At
the device locations, the structure is a composite in its thickness
direction: it is made up of two piezoelectric layers bonded
onto an elastic layer. Consequently, we consider a composite
beam [A, B] composed by two piezoelectric materials and one
elastic.
Assuming that the structure is composed by long beams,
Euler’s beam theory is used. The displacement in a cross
section of each beam of the structure is written as
U(x, y,z, t) = (u(x, t) + θ(x,t)z)x + v(x, t)z
where θ(x,t) = (∂v/∂x)(x, t), x is the local axis of the beam
and z the local axis in the beam thickness direction. The
element nodal displacement variable {U
e
} is then defined as
(figure 2)
{U
e
(t)}={
u
A
v
A
θ
A
u
B
v
B
θ
B
}
T
where (u
A
, v
A
, θ
A
) and (u
B
, v
B
, θ
B
) are the longitudinal
displacements, the normal displacements and the rotations
about the y-axis at nodes A and B. The generalized
displacements in the element can be expressed in nodal
variables by finite element interpolation functions as follows:
{U
h
}(x, t) ={
u
h
(x, t) v
h
(x, t) θ
h
y
(x, t)
}
T
= [N
e
(x)]{U
e
(t)}
where [N
e
] is the displacement shape functions matrix given
by
1 − x 00
01− 3x
2
+2x
3
L
e
(x − 2x
2
+ x
3
)
0
1
L
e
(−6x +6x
2
) 1 − 4x +3x
2
x 00
03x
2
− 2x
3
L
e
(−x
2
+ x
3
)
0
1
L
e
(6x − 6x
2
) −2x +3x
2
L
e
is the length of the finite element. The strain–displacement
is obtained using the strain differential operator:
{∂U
h
}(x, t) =
∂u
h
∂x
∂v
h
∂x
∂θ
h
y
∂x
T
(x, t)
= [B
e
U
]{U
e
(t)}.
For electric variables, as the thickness of piezoelectric
parts is small, we assume that the electric potential is
constant on each electrode and that the electric field is constant
in the piezoelectric; then they are directly associated with the
element and the element nodal electric potential variable {
e
}
is defined as (figure 2)
{
e
}={
1
2
3
4
}
T
where
1
,
2
,
3
,
4
represent the electric potentials on
each electrode of the element. These four element potential
variables allow us to consider many possible connections (in
order to excite bending or longitudinal motions) by coupling
in several ways the actuators or sensors. As the electric field
is constant, it is given by
{E
h
}=
E
(1)
E
(2)
=
−
1
h
1
1
h
1
00
00−
1
h
2
1
h
2
{
e
}
= [B
e
]{
e
}
where E
(1)
, h
1
and E
(2)
, h
2
are the electric fields in and the
thicknesses of piezoelectric parts 1 and 2.
Then, assuming that no electric charge is applied to
the piezoelectrics, variational principles give the following
element equations [5]:
[K
e
UU
]{U
e
} +[K
e
U
]{
e
} +[M
e
UU
]{
¨
U
e
}={F
e
U
} (1)
[K
e
U
]{U
e
} +[K
e
]{
e
}={0} (2)
where [K
e
UU
], [M
e
UU
] and {F
e
U
} are the element stiffness
matrix, the element mass matrix and the applied load vector.
These matrices contain homogenized mechanical coefficients
according to the beam section. Dots indicate a derivative with
respect to time. [K
e
U
] and [K
e
U
] couple the mechanical
properties to the electric properties and [K
e
] is the electric
stiffness matrix. For the piezoelectric layer i we define the
following constants:
S
(i)
=
A
(i)
z dy dz
(i)
U
=
d
(i)
31
E
(i)
S
(i)
h
(i)
(i)
U
=
e
(i)
311
S
(i)
h
(i)
(i)
U
=
d
(i)
31
E
(i)
A
(i)
h
(i)
(i)
U
=
e
(i)
311
A
(i)
h
(i)
(i)
=
(i)
33
A
(i)
L
e
(h
(i)
)
2
where d
(i)
31
, e
(i)
311
,
(i)
33
, E
(i)
, h
(i)
and A
(i)
are the piezoelectric
constants, Young’s modulus, the thickness and the section of
the piezoelectric layer, the expressions of [K
e
U
], [K
e
U
] and
[K
e
] are:
[K
e
U
] =
−
(1)
U
(1)
U
−
(2)
U
(2)
U
0000
(1)
U
−
(1)
U
(2)
U
−
(2)
U
(1)
U
−
(1)
U
(2)
U
−
(2)
U
0000
−
(1)
U
(1)
U
−
(2)
U
(2)
U
[K
e
U
] =
(1)
U
0 −
(1)
U
−
(1)
U
0
(1)
U
−
(1)
U
0
(1)
U
(1)
U
0 −
(1)
U
(2)
U
0 −
(2)
U
−
(2)
U
0
(2)
U
−
(2)
U
0
(2)
U
(2)
U
0 −
(2)
U
[K
e
] =
(1)
−
(1)
00
−
(1)
(1)
00
00
(2)
−
(2)
00−
(2)
(2)
.
Figure 2. Modeling of a composite beam.
The analytical expressions of [M
e
UU
] and [K
e
UU
] are detailed
in [6, 7].
Then, the assembled form of (1) and (2), for a beam
structure equipped with N
a
actuators and N
c
sensors can be
written as
[K
UU
]{q
U
} +[K
U
]
a
{q
}
a
+[M
UU
]{
¨
q
U
}={F
U
} (3)
[K
U
]
s
{q
U
} +[K
]
s
{q
}
s
={0} (4)
where {q
U
} (size Nddl), {q
}
s
(size
¯
N
s
) and {q
}
a
(size
¯
N
a
) are
the generalized displacements and the generalized potentials
of the sensors and the actuators.
¯
N
s
and
¯
N
a
are the number of
the unknown potentials of the sensors and actuators. They are
such that
¯
N
s
4N
s
and
¯
N
a
4N
a
.[K
UU
], [K
U
]
a
,[M
UU
],
[K
U
]
s
,[K
] and {F
U
} are the generalized discrete matrices.
In addition to these two equations, initial conditions have to be
added.
In order to set up a control law damping the vibrations
caused by external disturbances {F
U
} or by the initial
conditions, a state space model is developed in the next section
and a linear quadratic regulator (LQR) method, including a
state observer, is used.
3. The control system
The application of the active control methods in a dynamic
structural problem requires the use of a state space model. To
obtain this kind of equation, the solution {q
U
} is decomposed
into the normalized orthogonal modal basis {&
n
}. Assuming
that the contribution of the highest modes is negligible, we
keep only the first N eigenfunctions:
{q
U
}=
N
r=1
{&
r
}α
r
(t) = [&]
(Nddl,N)
{α}
(N,1)
. (5)
Substituting this equation into (3) and (4), and using the
orthogonality properties of modes leads to the state equations:
d{x}
dt
= [A]{x} +[B]{q
}
a
+ {g} (6)
{x}(t = 0) ={x
0
} (7)
and
{y}=[C]{x} (8)
where the normalized state vector (size 2N)is
{x}={
ω
n
α
n
˙α
n
}
T
(9)
[A], [B], [C] and {g} are the state, control output and load
matrices, given by:
[A] =
[0] [ω]
[−ω] −2[δ][ω]
[B] =
[0]
[&]
T
[K
U
]
a
[C] =
−[K
]
−1
s
[K
U
]
s
{&
r
}
ω
r
[0]
{g}=
{0}
[&]
T
{F }
{x
0
} is the initial conditions vector. As usual, a term of modal
viscous damping has been added to take into account a small
amount of damping. [δ] is the diagonal matrix of the damping
ratio and [ω] is the diagonal matrix containing the natural
angular frequencies.
In order to obtain a controlled system having good
stability and robustness, we chose the LQR control method [8].
Assuming that the state equation is controllable, it led us to use
the control law
{q
}
a
=−[K]{x} (10)
which minimizes a cost function given by
J
=
1
2
∞
0
[{x}
T
[Q]{x} + {q
}
T
a
[R]{q
}
a
]dt (11)
where [R] is a positive matrix and [Q] is a positive semidefinite
matrix. The choice of [Q] and [R] is not easy. In the following
applications, [Q] is chosen so that {x}
T
[Q]{x} represents the
mechanical energy. [R] is a diagonal matrix, the components
of which are chosen such that the maximal values of {q
}
a
are
less than the maximal admissible values for the piezoelectric
materials under consideration. In order to be implemented,
the optimal state control law obviously needs knowledge of the
state vector {x}. This knowledge is not complete since only the
output voltages in {y} are observed. Assuming that the state
system verifies the observability criteria, an estimation {ˆx} is
computed using a Luenberger observer [8, 9]. Consequently,
the control law applied to the actuators becomes
{q
}
a
=−[K]{ˆx}.
4. Applications
In this section, we present two applications. In each case
the structure is equipped with devices made with the same
piezoelectric material (figure 2). In order to limit the study
to bending motions, for each actuator and sensor we set
2
=
3
= 0, and
1
=−
4
(phase opposition between the
Table 1. Characteristics of the simple cantilever beam.
Length of the beam (m) 1
Length of the actuator 0.06
and the sensor (m)
Width (m) 0.02
Thickness (m) 0.002
Mass density (kg m
−3
) 2700
Young’s modulus (Pa) 7 × 10
10
Natural frequencies (Hz) 1.64, 10.29, 28.81, 56.46
Damping ratio 0.1%
Table 2. Characteristics of piezoelectric PZT.
Width (m) 0.01
Thickness (m) 0.001
Mass density (kg m
−3
) 7440
Young’s modulus (Pa) 4 × 10
10
Piezoelectric constant
33
1.72 × 10
−8
Piezoelectric constant d
31
(m V
−1
) 230 × 10
−12
Maximal admissible voltage (V) 250
Figure 3. A simple cantilever beam.
Figure 4. Comparison of the analytical calculus (black line) and
discrete model (grey line) for the sensor output.
two piezoelectric parts of each element). Consequently, here
we only consider one electric variable for each device. The
finite element was implemented in DYNADID2D [10]. The
construction of the control and the observer was done using
SCILAB [11].
First, we study the active control of a simple cantilever
beam in the case of a release test, equipped with one
piezoelectric actuator and one sensor located near the
fixed edge (figure 3). The geometrical and mechanical
characteristics of the system are detailed in tables 1 and 2. The
initial conditions are derived from an initial load
F(t = 0)
= 0.003z applied to the free end of the beam.
Because of the nature of the excitation, we take into
account only the first four eigenmodes. The study of this
simple structure can give us a first validation of the finite beam
element. The idea is to compare the results obtained using the
finite element discretization with analytical results [5]. For
this purpose the mechanical characteristics of piezoelectric
Table 3. Characteristics of the three-beam structure.
B
1
(m) (0, 0)
Length B
1
B
2
(m) 0.5
Length B
2
B
3
(m) 0.4
Length B
3
B
4
(m) 0.5
Location of actuator 1 (m) (0, 0.02)
Location of actuator 2 (m) (0.04, 0.5)
Location of sensor 1 (m) (0, 0.42)
Location of sensor 2 (m) (0.4, 0.46)
Length of each actuator (m) 0.06
Length of each sensor (m) 0.01
Natural frequencies (Hz) 1.48, 2.89, 7.99, 29.64
Width of elastic beams (m) 0.025
Thickness of elastic beams (m) 0.002
Mass density of elastic beams (kg m
−3
) 2700
Young’s modulus of elastic beams (Pa) 7.3 × 10
10
Damping ratio 0.025%
Figure 5. Comparison of the analytical calculus (black line) and
discrete model (grey line) for the actuator output.
Figure 6. Mechanical energy: release test under closed loop (black
line) and open loop (grey line) conditions.
are assumed to be negligible (in order to simplify analytical
developments). The output of the sensor and the required
input voltage obtained using the two methods (analytical and
discrete) are plotted in figures 4 and 5. For each figure, the
difference between the results cannot be seen as they are almost
identical: the finite element method, using the simple finite
beam element, gives the same results as the analytical calculus:
it is a first validation of the element.
We also studied the active control of the three-beam
structure shown figure 1, equipped with two actuators
