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Vibrations of an elastic structure with shunted
piezoelectric patches: ecient nite element formulation
and electromechanical coupling coecients
Olivier Thomas, Jean-François Deü, Julien Ducarne
To cite this version:
Olivier Thomas, Jean-François Deü, Julien Ducarne. Vibrations of an elastic structure with shunted
piezoelectric patches: ecient nite element formulation and electromechanical coupling coecients.
International Journal for Numerical Methods in Engineering, Wiley, 2009, 80 (2), pp.235-268.
�10.1002/nme.2632�. �hal-01572491�
Vibrations of an elastic structure with shunted piezoelectric
patches: efficient finite element formulation and
electromechanical coupling coefficients
O. Thomas,J.-F.Deü and J. Ducarne
Structural Mechanics and Coupled Systems Laboratory, Conservatoir
e National des Arts et M
´
etiers,
2 rue Conte
´
, 75003 Paris, France
This article is devoted to the numerical simulation of the vibrations of an elastic mechanical structure
equipped with several piezoelectric patches, with applications for the control, sensing and reduction of
vibrations. At first, a finite element formulation of the coupled electromechanical problem is introduced,
whose originality is that provided a set of non-restrictive assumptions, the system’s electrical state is fully
described by very few global discrete unknowns: only a couple of variables per piezoelectric patches,
namely (1) the electric charge contained in the electrodes and (2) the voltage between the electrodes. The
main advantages are (1) since the electrical state is fully discretized at the weak formulation step, any
standard (elastic only) finite element formulation can be easily modified to include the piezoelectric patches
(2) realistic electrical boundary conditions such that equipotentiality on the electrodes and prescribed
global charges naturally appear (3) the global charge/voltage variables are intrinsically adapted to include
any external electrical circuit into the electromechanical problem and to simulate shunted piezoelectric
patches. The second part of the article is devoted to the introduction of a reduced-order model (ROM) of the
problem, by means of a modal expansion. This leads to show that the classical efficient electromechanical
coupling factors (EEMCF) naturally appear as the main parameters that master the electromechanical
coupling in the ROM. Finally, all the above results are applied in the case of a cantilever beam whose
vibrations are reduced by means of a resonant shunt. A
finite element formulation of this problem is
described. It enables to compute the system EEMCF as well as its frequency
response, which are compared
with experimental results, showing an excellent agreement.
KEY WORDS
: piezoelectric materials; vibration reduction; electric boundary conditions; shunt;
electromechanical coupling
1
1. INTRODUCTION
Piezoelectric materials are proposed for many applications, most of the time to couple the vibra-
tions of a structure including the piezoelectric material with an electric circuit. This can lead to
several applications like oscillators for electronic circuits, acceleration sensors and gyroscopes,
sound transducers, vibration monitoring, control or energy harvesting and reduction of structural
vibrations. Both direct and indirect piezoelectric effects can be used, with the piezoelectric element
being used as sensors or actuators, or even both simultaneously. Except for very few applications
where the structure’s geometry—including the piezoelectric elements—is simple and allows an
analytical solution, numerical methods, such as the finite element method (FEM), have to be used
to compute the vibration response and simulate the system electromechanical behavior.
The present study proposes a finite element formulation adapted to elastic structures to which are
attached piezoelectric plate-like patches with electrodes. Its originality lies in the use of only one
pair of electrical variables per piezoelectric patch: the terminal voltage and the charge contained
in one of the electrodes. By essence, this choice of variable is naturally adapted to simulating
the behavior of the structure coupled to any electrical circuit and thus to practical engineering
applications.
A pioneering work on the subject of piezoelectric finite element [1] proposed a formulation of
the coupled problem between mechanical displacement and the electric potential field, taking into
account both direct and inverse effects. If the piezoelectric elements are used either as sensors
or actuators for active vibration control, it is worth remarking that the electrical unknowns can
be condensed so that the problem to solve has the form of a standard elastic vibration problem.
In particular, the actuators action on the system appears as an external forcing proportional to the
applied voltage, whereas the system’s stiffness is purely elastic, without any modifications due to
the electric state of the patches. In the case of sensors, the problem appears with an added stiffness
term due to the open-circuit condition of the piezoelectric patches, whose terminal voltage (the
sensor output) is proportional to the mechanical displacement unknowns. This formulation is the
basis of most numerical studies involving elastic structure with piezoelectric elements [2–10].For
more references, one can refer to the following review articles [11–13].
However, the need for modeling both sensing and actuation at the same time is required by
applications where a passive (or semi-passive) electric circuit is connected to the piezoelectric
elements. These so-called shunt techniques, proposed in [14], are reviewed in [15] in the case of
vibration reduction. The initial models used for simulating these shunted systems were lumped,
with only two degrees of freedom (d.o.f.) (one mechanical and one electrical, in most cases one
vibration mode and the electric-free charge). On the contrary, richer models can be obtained via
piezoelectric FEM commercial softwares, by simulating the shunt as an active controller with
simultaneous sensing and actuation [16]. This latter technique is often numerically intensive.
A new category of shunts has recently become of interest, often known as switch techniques,
where the electrical circuit impedance is switched periodically between two values [17–19].
In this case, the same distinction between the lumped model, characterized by fast computation,
and the full finite element simulations, more accurate but slower, are found [20]. A good compro-
mise is to use the modal basis to project the problem and reduce the number of variables without
losing the accuracy in the frequency domain [21].
In addition to simulating the vibratory behavior of the mechanical structure coupled to an electric
circuit, a key issue is the optimization of the whole system, in terms of size/shape/location of the
piezoelectric patches as well as the choice of the electric circuit components. In most cases, this
2
operation requires a maximization of the effecti
ve electromechanical coupling factors (EEMCF).
In the case of passive resistive or resonant shunt techniques, whose purpose is the structural
vibration reduction, the EEMCFs of the targeted vibration modes have to be maximized, as shown
in [14, 22, 23]. Moreover, it is proved in [24, 25] that the EEMCF is the only free parameter of
the optimization, the optimal value of the electric circuit parameters being known as functions of
the EEMCF and the system structural characteristics, and evaluated in a second step. The same
results are also obtained in the case of switch techniques [26].
The first goal of this article is to write a discretized formulation of the dynamics of an elastic
structure of arbitrary geometry with piezoelectric patches, the latter having a thin shape and being
polarized in their transverse direction. The main advantage of our formulation is that provided a
few non-restrictive electrical hypotheses, the system electrical state is defined by very few discrete
global unknowns: on the one hand, the free electric charges contained in the electrodes of the
piezoelectric patches and on the other hand the associated terminal potential differences. This
formulation enables to take into account both the direct and indirect piezoelectric effects, separately
or at the same time, while there is no need to compute the whole electric potential field. Concerning
the system mechanical state, no restrictions are formulated so that it appears in a standard finite
element form in the formulation, with the displacement d.o.fs at the mesh nodes and the associated
generalized forces.
The present problem formulation can be used to simulate the dynamics of the system in any
standard situation of a smart structure connected to an electrical network: a part of the piezoelectric
patches can be sensors and the remaining part actuators. However, this study especially addresses
the simulation of the dynamics of the system coupled to shunted electric circuits, since each
piezoelectric patch appears in the formulation with a pair of global intuitive unknowns, its potential
difference and its electric charge, that can be introduced without restrictions in the evolution
equation of the electrical circuit to which it is plugged. This is an originality of this work, since
most other studies are restricted to piezoelectric elements used either as sensor or as actuators (see
above cited references). Another advantage of our formulation is that realistic electric boundary
conditions on the piezoelectric patches—i.e. on the one hand equipotentiality in the electrodes
and on the other hand a global free electric charge in the electrodes—are naturally imposed and
included in our formulation. This is another originality since classical finite element formulations
of the literature often use local electric variables—the electric potential and displacement—that
are not naturally adapted to the above cited global boundary conditions. The equipotentiality in
the electrodes is often enforced by assigning a single d.o.f. for voltage on all their nodes [27–29].
The second goal of this work is to introduce an efficient manner of evaluating the effective
electromechanical coupling factors (EEMCF) of the system. As stated above, most of the literature
works on the subject show that the EEMCFs are the major parameters that govern, on the one
hand, all the performances of the vibration dampers made with shunted piezoelectric elements,
and on the other hand, all the optimizations procedures of those devices [24–26]. Thus, evaluating
the EEMCF for a given system is crucial when designing its coupling with an electrical circuit.
To do so, a given EEMCF is associated in our study with one system vibration eigenmode and
one piezoelectric patch. The general discretized finite element formulation introduced above is
expanded onto the vibration eigenmodes of the system with all its piezoelectric elements short-
circuited. A modal discretized formulation is obtained, where all the EEMCFs appear naturally.
The latter are known with analytical expressions as functions of the matrices of the finite element
discretization and the short-circuit eigenmodes. This result can be useful in an engineering context,
since the exhibited EEMCFs numerical computation involves only the assembling of the finite
3
element matrices and the computation of the system short-circuit eigenmodes, which take the form
of a standard elastic eigenvalue problem. To conclude, the EEMCFs for a given elastic structure
with piezoelectric patches can potentially be computed in post-processing, after a modal analysis
done with a standard finite element elastic code.
The present article is divided into six main parts. After the present introduction, Section 2
recalls the main equation of a standard piezoelectric vibration problem and the associated varia-
tional formulation. Section 3 gives the main electrical hypotheses that lead to introduce the global
electrical variables—the voltage/charge pair associated with each piezoelectric patch—in the vari-
ational formulation and derives the associated general discretized finite element formulation. The
modal expansion of the general finite element formulation is proposed in Section 4, thus, enabling
to introduce and compute the problem EEMCFs. Section 5 exposes the details of a finite element
discretization of a laminated beam with piezoelectric patches. To conclude, Section 6 applies all
the above formulations to a beam with two collocated piezoelectric patches connected to a resonant
shunt circuit. The EEMCFs are numerically evaluated as well as the vibratory response of the beam
with and without the shunt. Experiments are described and an excellent agreement is obtained,
thus, validating the introduced formulations.
2. AN ARBITRARY PIEZOELECTRIC MEDIUM
This section is devoted to the general formulation of the equations that govern the mechanical
and electric state of an arbitrary piezoelectric medium, on their strong form as well as on a weak
form, suitable for a finite element formulation. Standard indicial notations are adopted throughout
the article: subscripts i, j,k,l denote the three-dimensional vectors and tensor components and
repeated subscripts imply summation. In addition, a comma indicates a partial derivative.
2.1. Electro-mechanical local equations
We consider a piezoelectric structure occupying a domain
p
at the equilibrium. The structure
is subjected to a prescribed displacement u
d
i
on a part
u
and to a prescribed surface force
density t
d
i
on the complementary part
t
of its external boundary. The electric boundary conditions
are defined by a prescribed electric potential
d
on
and a surface density of free electric
charges q
d
on the remaining part
q
. Thus, the total structure boundary, denoted *
p
,issuch
that *
p
=
u
∪
t
=
∪
q
with
u
∩
t
=
∩
q
=∅. In addition,
p
is subjected to prescribed
body forces f
d
i
.
The linearized deformation tensor and the stress tensor are denoted, respectively, by ε
ij
and
ij
.
Moreover, D
i
denotes the electric displacement vector components and E
i
the electric field vector
components. is the mass density, n
i
is the unit normal external to
p
and t is the time.
The local equations of the electro-mechanical coupled problem are:
ij, j
+ f
d
i
=
*
2
u
i
*t
2
in
p
(1a)
ij
n
j
=t
d
i
on
t
(1b)
u
i
=u
d
i
on
u
(1c)
4
