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Hybrid Active-Passive Damping Treatments Using
Viscoelastic and Piezoelectric Materials: Review and
Assessment
MARCELO A. TRINDADE
Department of Mechanical Engineering, Pontifıcia Universidade Catolica do Rio de Janeiro, rua
Marques de Sao Vicente, 225, 22453-900 Rio de Janeiro, Brazil
AYECH BENJEDDOU
StructuralMechanics andCoupled Systems Laboratory, Conservatoire National des Arts etMetiers,
2 rue Conte, 75003 Paris, France
Abstract: Hybrid active-passive damping treatments combine the reliability, low cost and robustness of vis-
coelastic damping treatments and the high performance, modal selective and adaptive piezoelectric active
control. Numerous hybrid damping treatments have been reported in the literature. They differ mainly by
the relative positions of viscoelastic treatments, sensors and piezoelectric actuators. Therefore, the present
article provides a review of the open literature concerning geometric configurations, modeling approaches
and control algorithms for hybrid active (piezoelectric)-passive (viscoelastic) damping treatments of beams.
In addition, using a unified finite element model able to represent sandwich damped beams with piezoelec-
tric laminated faces and an optimal control algorithm, the geometric optimization of four hybrid treatments
is studied through treatment length and viscoelastic material thickness parametric analyses. A comparison
of the performances of these hybrid damping treatments is carried out and the advantages and drawbacks of
each treatment are identified. Beside the literature review of more than 80 papers, the present assessment has
the merit to present for the first time detailed parametric and comparative analyses for these already known
hybrid active (piezoelectric)-passive (viscoelastic) damping configurations. This may be of valuable help for
researchers and designers interested in this still growing field of hybrid active-passive damping systems.
Key Words: Vibration control, hybrid active-passive damping treatments, viscoelastic materials, piezoelectric mate-rials
NOMENCLATURE
AC = Active Control
AC/PCL = Active Control / Passive Constrained Layer
AC/PSOL = Active Control / Passive Stand-Off Layer
ACL = Active Constrained Layer
ADF = Anelastic Displacement Fields
APCL = Active-Passive Constrained Layer
ATF = Augmenting Thermodynamic Fields
CM = Complex Modulus
D = Derivative
Journal of Vibration and Control, 8: 699–745, 2002 DOI: 10.1177/1077546029186c©2002 Sage Publications
700 M. A. TRINDADE and A. BENJEDDOU
dof = Degrees of freedom
DVF = Direct Velocity Feedback
EACL = Enhanced Active Constrained Layer
FE = Finite Element
FRF = Frequency Response Function
GHM = Golla-Hughes-McTavish
LQG = Linear Quadratic Gaussian
LQR = Linear Quadratic Regulator
MSE = Modal Strain Energy
P = Proportional
PCL = Passive Constrained Layer
PD = Proportional-Derivative
PPF = Positive Position Feedback
PSOL = Passive Stand-Off Layer
PWM = Progressive Waves Method
1. INTRODUCTION
The use of piezoelectric materials for sensing and control of flexible structures has been well
studied in the past two decades (Sunar and Rao, 1999). Indeed, these materials are well
adapted to distributed structural vibration sensing and control since very thin piezoelectric
layers or patches can be bonded or embedded in host structures. For the active control of small
amplitude vibrations of very flexible structures, they lead to lightweight, adaptive and high
precision and performance control systems (Sunar and Rao, 1999). However, it is well known
that active controllers are very sensitive to variations and uncertainties of system parameters.
Moreover, the most used piezoceramic materials are very brittle, so that they can fail under
operation. Hence, adding some passive damping to the structure may lead to more reliable
and robust performances.
On the other hand, purely passive damping treatments generally lead to reliable, low cost
and robust vibration control (Mead, 1999). These can be achieved by covering part of the
structure with constrained or unconstrained layers of viscoelastic materials. Several polymers
exhibit the viscoelastic properties of naturally dissipating vibratory energy into heat energy.
This is why they are largely used in the aerospace, aeronautical and automotive industries
to provide passive vibration damping. Nevertheless, the efficiency of such treatments is
dependent on the volume of material used so that their performance is generally limited by
weight and size constraints. Hence, adding active vibration control may improve damping
while respecting structural constraints.
In recent years, researches have been directed to simultaneous use of piezoelectric and
viscoelastic materials to provide reliable, robust, adaptive and effective damping treatments
(Benjeddou, 2001; Inman and Lam, 1997; Lesieutre and Lee, 1996). Depending on the relat-
ive positions of the viscoelastic layer and the piezoelectric actuator, the viscoelastic passive
and piezoelectric active actions can operate either separately or simultaneously. However,
most of the research in this area has been focused on simultaneous actions (Inman and Lam,
1997). In fact, only recently separate active and passive control mechanisms have been
analyzed.
The relatively large number of papers in the area of hybrid active-passive damping is
due to the high potential of industrial applications and the multidisciplinary questions raised
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 701
by such systems. In fact, these multi-physics systems evolve some complexities that can be
listed as:
• Modeling of laminated piezoelectric structures due to the electromechanical coupling in-
troduced by the piezoelectric sensors and actuators bonded on or embedded in the structure
(Benjeddou, 2000; Saravanos and Heyliger, 1999);
• Providing realistic models of viscoelastic materials, since their properties vary with oper-
ating temperature and frequency, amplitude and type of excitation (Mead, 1999);
• Development of active control algorithms well adapted for damped structures and, if pos-
sible, taking advantage of the passive damping mechanism.
Consequently, the objective of the present article is, first, to provide a review
of the published works concerning geometric configurations, modeling approaches and
control algorithms for hybrid active (piezoelectric)-passive (viscoelastic) damping treatments
of beams. Then the main ones are assessed through geometric optimization of each
configuration and comparison of their performances, both optimal and overall, in order to
draw some conclusions about each treatment advantages and drawbacks.
2. LITERATURE REVIEW
The development of hybrid active-passive vibration control systems has been the object of
several recent research projects. The analysis of the publications found in the open literature
shows that the main centers of interest have been: (1) configurations of hybrid damping
treatments; (2) modeling of sandwich/multilayer structures; (3) modeling of viscoelastic
materials and (4) active control algorithms. In the following sub-sections, a bibliographic
review concerning each one of these topics is presented. Since hybrid active-passive damping
treatments have been mostly applied to beams, the following literature analysis is limited to
these structures. Readers are invited to see Benjeddou (2001) for other structural elements,
such as plates and shells.
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Damping treatments using a Passive Constrained viscoelastic Layer (PCL) are already largely
used to damp out structural vibrations (Mead, 1999). For some applications, aiming at an
integrated damping, it is also possible to replace parts of the structure by others in the form
of a sandwich with viscoelastic cores, or even to conceive sandwich parts directly integrated
in the structure. On the other hand, piezoelectric materials may be used as actuators, through
their reverse piezoelectric effect, that, associated to analogic or digital controllers, allows
the provision of active control (AC) of structural vibrations. The success of these passive
treatments combined with the progress obtained in the field of smart structures motivated
the development of hybrid active-passive damping treatments, consisting in augmenting or
replacing the elastic constraining layer by piezoelectric actuators. These types of treatment,
so-called Active Constrained Layer (ACL) damping, was first suggested by Plump and
Hubbard Jr. (1986).1 Their main advantages lie in (i) their capacity to increase actively the
shear strains in the viscoelastic material, through the piezoelectric actuator, thus improving
energy dissipation; (ii) combination of the performances at higher frequencies of viscoelastic
702 M. A. TRINDADE and A. BENJEDDOU
treatments and, at very low frequencies, of the active control through piezoelectric actuation;
(iii) increase in the robustness of active control, insofar as, in the event of operation fail, the
system remains passively damped.
Since the beginning of the 1990s, several configurations of ACL treatments have been
reported in the literature. That proposed by Plump and Hubbard Jr. (1986) was retained by
the majority of the researchers. Nevertheless, a multitude of hybrid damping treatments was
proposed, according to the relative positions of viscoelastic and piezoelectric layers and the
position and type of sensors and actuators. Figure 1 shows some of the main hybrid damping
treatments configurations found in the literature. Using the standard ACL configuration
of Plump and Hubbard Jr. (1986), Shen (1994; 1996) and Liao and Wang (1997a) used
optical sensors to measure the beam tip deflection (Figure 1(a)), which was processed and
transmitted to the piezoelectric actuator to provide damping enhancement. Varadan, Lim, and
Varadan (1996) considered a piezoelectric sensor bonded beside the hybrid treatment (Figure
1(b)). Using measurements of the beam bottom surface strains, Lesieutre and Lee (1996)
proposed segmenting the ACL treatment (Figure 1(c)) in order to increase its robustness,
although the performances for the higher frequencies were decreased. Trindade, Benjeddou,
and Ohayon (2000b) also used segmented ACL treatments to provide additional hybrid
damping performance. This configuration was also extended by Baz (1997c) by considering
two treatments bonded symmetrically on the surfaces of a beam, but only longitudinal
displacements were considered in order to study the mechanism of shear enhancement in
the viscoelastic layer. Agnes and Napolitano (1993), Huang, Inman, and Austin (1996) and
Yellin and Shen (1996) considered self-sensing actuators in the active constraining layer (Fig-
ure 1(d)) to simultaneously actuate and sense the structural vibrations. This has the advantage
of leading to a collocated control system (Dosch, Inman, and Garcia, 1992).
A variation of the preceding configuration was proposed by Baz (1998) and Baz and Ro
(1993; 1995a) by placing a thin piezoelectric polymer sensor between the viscoelastic layer
and the structure (Figure 1(e)). This configuration leads to a good collocalization between the
actuator and sensor without having to use the complex circuits of the self-sensing actuator.
This treatment was also retained by Veley and Rao (1996) and, in segmented version, by
Kapadia and Kawiecki (1997) (Figure 1(f)).
In order to actively treat structures already covered by PCLs, Azvine et al. (1993)
proposed to bond a piezoelectric patch on the existing PCL (Figure 1(g)). This configuration
will be named here as Active-Passive Constrained Layer (APCL) damping since the
constraining layer is composed of passive and active sub-layers. This configuration was also
considered in Azvine, Tomlinson, and Wyne (1995) and Rongong et al. (1997) but with two
treatments bonded symmetrically on the top and bottom surfaces of the beam (Figure 1(h)).
Measurements of the beam tip displacements are taken, using an accelerometer for the first
case and an optical sensor for the other. This construction presents the advantage of allowing
the lengths of the active and passive treatments to be different, in contrast to the preceding
ACL ones which require actuators entirely covering the viscoelastic layers.
According to theworks cited above, hybrid damping treatments are alwaysmore effective
than passive ones. Compared to purely active control, they are more effective for short
treatments (Huang, Inman, and Austin, 1996), relatively low control gains (Huang, Inman,
and Austin, 1996; Liao and Wang, 1997a), rigid viscoelastic materials (Liao and Wang,
1997a), and more robust (Lesieutre and Lee, 1996; Rongong et al., 1997), since the stability
margin of the modes excited by the controller is augmented. From another point of view,
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 703
Figure 1. Hybrid active-passive damping configurations.
704 M. A. TRINDADE and A. BENJEDDOU
hybrid damping treatments make it possible to have a performance comparable to those
of active and passive treatments alone, with lighter treatments (less viscoelastic material)
(Veley and Rao, 1996) and less control power (lower control voltages) (Liao and Wang,
1997a). However, it was found that these hybrid treatments suffer from loss of transmissibility
between the piezoelectric actuator and the structure, due to the flexibility of the viscoelastic
layer (Lam, Inman, and Saunders, 1997; Liao and Wang, 1996).
To remedy the transmissibility reduction problem, other hybrid damping configurations
were proposed in the literature. Hence, Liao and Wang (1996) suggested adding rigid
elements at the edges of the actuators to connect them to the structure and, therefore, lead to a
certain direct action of the actuators (Figure 1(i)). Although this treatment, named Enhanced
ACL (EACL), allows, indeed, an increase in the transmissibility between the actuator and
the structure, the shear strain in the viscoelastic layer is reduced and the treatment thus
becomes less passively damped. According to Liao and Wang (1996), the edge elements
widen the regions of optimality of ACL treatments, compared to purely passive and active
treatments. This configurationwas also studied byVaradan, Lim, andVaradan (1996) in order
to determine optimal forms for the edge elements. They considered a piezoelectric sensor
beside the treatment (Figure 1(j)), whereas Liao and Wang (1996) measured the deflection of
the beam with an optical sensor. Liu and Wang (1998) also considered the EACL treatment
but using a self-sensing actuator (Figure 1(k)). Badre-Alam, Wang, and Gandhi (1999) have
considered two EACL treatments bonded symmetrically on the opposite surfaces of the beam
(Figure 1(l)).
Another way of solving the problem of the actuator-structure transmissibility reduction
is to consider active and passive treatments acting separately. Thus, Chen and Baz (1996)
proposed bonding a piezoelectric actuator directly on the bottom surface of a beam, in
addition to anACL treatment bonded on the opposite surface (Figure 1(m)). This construction
was considered also by Crassidis, Baz, and Wereley (2000) in order to compare the
performances obtained while acting through the actuator of the ACL treatment, on one
hand, and through that on the opposite side of the beam, on the other hand. In the last
case, the ACL actuator over the viscoelastic layer acts passively like a standard PCL. Their
results showed that ACL is more effective and requires lower actuation voltages than the
combination of purely active control with a passive constrained layer. Other researchers
had also studied separated active and passive treatments, like Lam, Inman, and Saunders
(1997), who presented separated active (AC) and passive (PCL) treatments, bonded on the
same side (Figure 1(n)) and on the opposite sides (Figure 1(o)) of a beam, leading to the so-
called AC/PCL treatments. Their results showed that, for the studied case, the two AC/PCL
treatments are more effective and require less control voltages than the ACL one, and that
AC/PCLs are more effective when active and passive treatments are placed on opposite sides.
Friswell and Inman (1998) considered also a beam treated by a PCL, on its top surface, and by
a self-sensing piezoelectric actuator, on the opposite surface (Figure 1(p)). Later, Lam, Inman,
and Saunders (1998) suggested placing the piezoelectric actuator underneath the viscoelastic
material (Figure 1(q)). It was found that the last configuration leads to better passive damping.
In fact, placing the actuator underneath the PCL treatment allows simultaneously to have
direct AC action and elevate the PCL treatment, which yields a Passive Stand-Off Layer
(PSOL) damping (Yellin and Shen, 1998). That is why this treatment is differentiated from
the AC/PCL ones (Figures 1(n) and 1(o)) and named AC/PSOL damping treatment. A list of
papers for each hybrid configuration discussed in this section is given in Table 1.
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 705
Table 1. Hybrid damping treatments configurations used in the literature.
ACL Figure 1(a)–(f) Agnes and Napolitano (1993), Baz (1997c; 1998), Baz
and Ro (1993; 1995a), Huang, Inman, and Austin (1996),
Kapadia and Kawiecki (1997), Lesieutre and Lee (1996),
Liao and Wang (1997a), Plump and Hubbard Jr. (1986),
Shen (1994; 1996), Trindade, Benjeddou, and Ohayon
(2000a; 2000b), Varadan, Lim, and Varadan (1996), Yellin
and Shen (1996)
APCL Figure 1(g),(h) Azvine et al. (1995; 1993), Rongong et al. (1997)
EACL Figure 1(i)–(l) Badre-Alam, Wang, and Gandhi (1999), Liao and Wang
(1996; 1998a; 1998b), Liu and Wang (1998), Varadan, Lim,
and Varadan (1996)
AC/PCL Figure 1(m)–(p) Chen and Baz (1996), Crassidis, Baz, and Wereley (2000),
Friswell and Inman (1998), Lam, Inman, and Saunders (1997)
AC/PSOL Figure 1(q) Lam, Inman, and Saunders (1997; 1998)
As shown in Figure 1, the majority of the hybrid damping treatment configurations are
made up of more than three layers. Consequently, multilayer beam models should be used,
although sandwich beam models were generally adapted through simplifying assumptions.
The ACL, APCL, AC/PCL and AC/PSOL hybrid treatments will be analyzed later in this
paper through a parametric analysis using a laminated faces sandwich beam model able to
deal with these different configurations.
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The first study on the vibration of sandwich beams with viscoelastic cores was presented
by Kerwin (1959) towards the end of the 1950s. He considered simply supported sandwich
beams, whose elastic surface layers were much more rigid than the core and whose bending
rigidity of the constraining layer was negligible compared to that of the base structure. Thus,
the core was supposed to deform only in transverse shear and the bending rigidity of the
sandwich beamwas that of the base structure. DiTaranto (1965) extended the work of Kerwin
to treat the free vibrations of sandwich beams with arbitrary boundary conditions. Then,
while keeping the assumptions of Kerwin (1959), another model was developed by Mead
and Markus (1969) to study the forced vibrations for several boundary conditions leading to
a differential equation of motion of sixth order in the transverse deflection.
These first sandwich beammodels were, recently, extended to the case of ACL treatments
to account for a piezoelectric constraining layer. Hence, Agnes and Napolitano (1993) used
the theory of Kerwin (1959), considering the piezoelectric actuator effect as an increase in
the stiffness of the passive constraining layer. Leibowitz and Vinson (1993) have studied
the same problem, but for a beam partially covered by the ACL. In addition, Baz and Ro
(1993) started from the equation of Mead and Markus (1969) to analyze by the Progressive
Waves Method (PWM) a cantilever beam partially treated by an ACL, although in their case
a piezoelectric sensor was placed between the viscoelastic material and the beam. That is
why, in order to remain within the framework of sandwich beam theory, they supposed that
the sensor and the beam formed a single layer whose rigidity was equivalent to the sum of
those of the two layers. In this case, the piezoelectric action was simplified by an imposed
706 M. A. TRINDADE and A. BENJEDDOU
βcz
w’
w
ua
uc
ub
x
Viscoelastic layer
Elastic/piezoelectric layers
Elastic/piezoelectric layers
Figure 2. Laminated faces sandwich beammodel description (Trindade, Benjeddou, andOhayon, 2001a).
strain. This was also supposed by Shen (1994) in order to derive the equations of motion of a
beam completely covered by an ACL treatment which was then analyzed through the transfer
functions approach.
To study three-layers sandwich beams, van Nostrand and Inman (1995) developed a finite
element (FE) model with four degrees of freedom (dof) per node, for which, just like Baz
(1998), Huang, Inman, and Austin (1996) and Liao and Wang (1998a), they supposed only
transverse shear strain in the viscoelastic layer. A similar FE model was developed by Baz
and Ro (1995b) for their ACL configuration (Figure 1(e)), but accounting also for extension
and bending of the viscoelastic layer. Then, Lesieutre and Lee (1996) developed a FE model,
adding the effects of rotational inertia for the three layers, with quadratic interpolations of
the longitudinal displacement and the shear angle, leading to 9 dof per element. In addition,
isoparametric finite elements were used to study EACL and ACL treatments by Varadan,
Lim, and Varadan (1996) and ACL treatments by Veley and Rao (1996). In both cases,
electric dof were used to model the piezoelectric effect. In order to model ACL treatments,
Trindade, Benjeddou, and Ohayon (2000b) used a sandwich beam model with 8 dof per
element, proposed by the same authors (Benjeddou, Trindade, and Ohayon, 1999), including
bending and extension of the viscoelastic layer and rotational inertia of all layers.
The Rayleigh-Ritz method was also used to discretize the equations of motion by Lam,
Inman, and Saunders (1997) and Liao and Wang (1997a), where the rotational inertia was
neglected but the axial strains of the viscoelastic layer were considered. The last model (Liao
and Wang, 1997a) was used by the same authors to study the effect of the edge elements
(Figure 1(i)) in Liao and Wang (1996), the latter being modeled by springs connecting the
actuator to the beam.
Beams with more than three layers were also treated in the literature. However, as
proposed in Trindade, Benjeddou, and Ohayon (2001a) and shown in Figure 2, a three-layer
sandwich beammodel may be used, by supposing that surface layers are composed of several
sub-layers. Then, the same displacement fields (1) of a surface layer are supposed for all of
its sub-layers. In Trindade, Benjeddou, and Ohayon (2001a), only the core is supposed to
undergo shear strains (Timoshenko theory) while the surface layers are assumed to respect
Bernoulli-Euler theory,
uk (x, y, z) = uk (x)− (z− zk )w′(x), k = a, b
uc(x, y, z) = uc(x) + zβ c(x)
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 707
wi (x, y, z) = w(x), i = a, b, c (1)
Chen and Baz (1996), and Crassidis, Baz, and Wereley (2000) extended the work of Baz
and Ro (1993; 1995b) to study a beam treated with a three-layer ACL treatment (Baz and Ro,
1993) bonded on its upper surface, and a piezoelectric actuator bonded on the opposite surface
(Figure 1(m)). In both cases (Chen and Baz, 1996; Crassidis, Baz, and Wereley, 2000), the
beam and the sensor of the opposite surface were supposed to compose a single layer. In the
work of Chen and Baz (1996), the stiffness of the piezoelectric sensor was also neglected.
A FE model of a beam with five layers (Figure 1(l)) was also developed by Badre-Alam,
Wang, and Gandhi (1999) in order to study two ACL treatments with edge elements (EACL
(Liao and Wang, 1996)) bonded on the upper and lower surfaces of a beam. In this case,
the viscoelastic material was supposed to deform in transverse shear only. Electric dof were
considered for the piezoelectric layers, leading to a finite element with 7 dof per node.
In order to study a sandwich beam with viscoelastic core and two piezoelectric actuators
bonded on its upper and lower surfaces,Wang andWereley (1998) used themodel ofMead and
Markus (1969) associated with the PWM. However, the piezoelectric actuators were replaced
by concentrated moments at the edges, thus reducing the model to that of a three layer beam.
The sandwich beam of Wang and Wereley (1998) was also studied by Trindade, Benjeddou,
and Ohayon (2001a) but modeling also the piezoelectric layers through a FE model with 4
mechanical dof/node after condensing the electric dof. They then used this model to analyze
its hybrid damping performance. In the present study, the finite element model presented in
Trindade, Benjeddou, and Ohayon (2001a) is used since it is devoted to sandwich beams with
laminated elastic/piezoelectric surface layers and thus is able to well represent beams treated
by ACL, APCL, AC/PCL and AC/PSOL hybrid damping configurations.
Representing the model with the greatest number of layers among those found in the
literature, that proposed by Rongong et al. (1997) (Figure 1(h)) considered a beam treated
on its upper and lower surfaces by APCL treatments with three layers, generalizing that of
Azvine et al. (1993) (Figure 1(g)). Nevertheless, the beam, the viscoelastic layer and the
piezoelectric and constraining layers were supposed to deform, respectively, in bending, shear
and extension only. Moreover, the constraining layer and the piezoelectric actuator were
considered as a single layer, leading to a symmetrical five-layer beam. The Rayleigh-Ritz
method was used in this case to discretize the equations of motion.
One can summarize the main assumptions used in the previously cited works as follows:
1. Pure shear in the viscoelastic layer, except for Baz and Ro (1995b), Crassidis, Baz, and
Wereley (2000), Lam, Inman, and Saunders (1997), Lesieutre and Lee (1996), Liao and
Wang (1996; 1997a), Trindade, Benjeddou, andOhayon (2000a; 2000b; 2001a), Varadan,
Lim, and Varadan (1996) and Veley and Rao (1996);
2. Negligible rotational inertia, except for Agnes and Napolitano (1993), Lesieutre and Lee
(1996), Trindade, Benjeddou, and Ohayon (2000a; 2000b; 2001a), Varadan, Lim, and
Varadan (1996) and Veley and Rao (1996);
3. Negligible shear in the elastic and piezoelectric layers, except for the isoparametric FE
models of Varadan, Lim, and Varadan (1996) and Veley and Rao (1996);
4. Piezoelectric actuator modeled using induced strain, force or stiffness, except for Badre-
Alam, Wang, and Gandhi (1999), Lam, Inman, and Saunders (1997), Liao and Wang
(1996; 1997a), Trindade, Benjeddou, and Ohayon (2000a; 2000b; 2001a), Varadan, Lim,
and Varadan (1996), Veley and Rao (1996).
708 M. A. TRINDADE and A. BENJEDDOU
The majority of the works cited in this sub-section supposed sinusoidal vibrations,
which enabled them to consider a complex shear modulus to account for elastic and viscous
properties of viscoelastic materials. However, this approach is not generally realistic.
Moreover, the hybrid treatment performance is strongly dependent on the behavior of the
viscoelastic material; therefore, its modeling is very important.
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The elastic and dissipative properties of viscoelastic materials depend, generally, on the
frequency, operating temperature, amplitude and type of excitation. The need for viscoelastic
damping models able to represent the physical reality of these properties motivated numerous
studies in the 1980s. At that time, the complex modulus (CM) approach was already largely
used, in particular by major early papers on hybrid damping treatments. In this method, the
shear modulus of the viscoelastic material is written as
G ∗(ω) = G ′(ω)[1 + iη(ω)] (2)
where G ′(ω) and η(ω) are the storage modulus and loss factor, respectively. Although
these are frequency-dependent, a constant version of this formula was often retained (Azvine,
Tomlinson, and Wynne, 1995; Baz, 1997a; 1997b; 1997c; Shen, 1994; 1997) for simplicity.
To account for frequency dependence, the CM exponential formula of Douglas and Yang
(1978) was considered in Huang, Inman, and Austin (1996), Shen (1995; 1996), and
Varadan, Lim, and Varadan (1996). This formula supposes constant loss factor and
frequency-dependent storage modulus such that
G ∗(ω) = 0.142( ω
2π
)0.494
(1 + 1.46i)MPa. (3)
A frequency-by-frequency analysis using the shear modulus master curves was
sometimes preferred (Agnes and Napolitano, 1993; Baz, 1998; Crassidis, Baz, and Wereley,
2000; Rongong et al., 1997). In this case, both storage modulus and loss factor are frequency
dependent. For representation of the frequency-dependent viscoelastic material properties,
the PWM has been also used (Baz and Ro, 1993; Wang and Wereley, 1998) for frequency-
domain formulation.
The CM approach is mostly associated to the so-called Modal Strain Energy (MSE)
method proposed by Johnson, Keinholz, and Rogers (1981), which states that the rth structuremodal loss factor η r
s may be evaluated by
η rs = η
H rv
H rs
(4)
where H rs and H
rv are the strain energies, associated with the rth mode, of the structure and
the viscoelastic material, respectively. This method is known to lead to good viscoelastic
damping estimation for low damping only. Nevertheless, since it considers that the visco-
elastic material properties remain constant for all loading conditions, its use is generally
limited to sinusoidal loads. Thus, to account for the frequency dependence of the viscoelastic
material properties, this method should be considered in an iterative version (Friswell and
Inman, 1998).
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 709
Table 2. Some mathematical models for the viscoelastic relaxation function h(s).
Relaxation function h(s)/G0 Reference∑i
ais+ bi
(Biot, 1955)
e1sα − e0bsβ
1 + bsβ,
0 < α < 10 < β < 1
(Bagley and Torvik, 1983)
∑i
∆i s
s+Ωi
(Lesieutre, 1992)
∑iα i
s2 + 2ζ i ωi s
s2 + 2ζ i ωi s+ ω2
i
(McTavish and Hughes, 1993)
τ 0s+∑
i
α iτ i s
τ i s+ 1(Yiu, 1993)
Several time-domain methods able to represent the frequency dependence of viscoelastic
materials properties and compatible with the analysis techniques generally used in structural
dynamics have been also proposed in the literature. These are based on the theory of linear
viscoelasticity, from which the jth component of the constitutive equations of a viscoelastic
material initially at rest is written as
σj (t) = G(t)ε j (0) +
∫ t
0
G(t− τ )∂εj∂τ
(τ ) dτ . (5)
For nil initial conditions, (5) may be Laplace-transformed to
σ j (s) = [G0 + h(s)]ε j (s) (6)
whereG0 represents the stiffness of the material and h(s) its relaxation or dissipation. Severalmathematical models have been proposed for the latter function (Table 2). Thus, Bagley and
Torvik (1983) proposed a method effective for frequency-domain analyses of viscoelastic
damping but leading, in the time-domain, to fractional-order differential equations rather
complicated to solve. Other time-domain methods were then proposed yielding to ordinary
differential equations. Lesieutre (Lesieutre, 1992; Lesieutre and Mingori, 1990) proposed a
model named Augmenting Thermodynamic Fields (ATF), based on the introduction of internal
(dissipative) variables, similar to the hidden variables introduced by Biot (1955) in his linear
theory of irreversible thermodynamics. Limited to unidimensional cases, this model was later
extended to three-dimensional ones by Lesieutre and co-workers (Lesieutre and Bianchini,
1995; Lesieutre and Lee, 1996) through a new model named Anelastic Displacement Fields
(ADF). In parallel, Dovstam (1995) presented a similar study, although only a frequency-
domain analysis was carried out. Hughes and co-workers (Golla andHughes, 1985; McTavish
and Hughes, 1993) also developed a model, the so-called Golla-Hughes-McTavish (GHM)
model, also based on the introduction of dissipative variables.
The ATF/ADF and GHMmodels are rather similar since they all use additional variables
to model viscoelastic material relaxation. Nevertheless, one can distinguish some differences
between them. The GHM model uses a Laplace-domain formulation, thus requiring it to be
retransformed to time-domain space. However, it leads to a second order equation compatible
710 M. A. TRINDADE and A. BENJEDDOU
Table 3. Viscoelastic damping models used in the literature of hybrid damping.
Complex modulus Azvine, Tomlinson, and Wynne (1995), Baz (1997a;
1997b; 1997c), Shen (1994; 1997)
Iterative complex modulus Agnes and Napolitano (1993), Baz (1998), Baz and
Ro (1993), Crassidis, Baz, and Wereley (2000),
Rongong et al. (1997), Wang and Wereley (1998)
Formula of Douglas and Yang Huang, Inman, and Austin (1996), Shen (1995; 1996),
(1978) Varadan, Lim, and Varadan (1996)
Modal Strain Energy Veley and Rao (1996)
Iterative Modal Strain Energy Friswell and Inman (1998), Trindade, Benjeddou, and
Ohayon (2000a; 2000b)
Golla-Hughes-McTavish Badre-Alam, Wang, and Gandhi (1999), Friswell,
Inman, and Lam (1997), Lam, Inman, and Saunders
(1997), Liao and Wang (1996; 1997a; 1997b; 1998b),
Trindade, Benjeddou, and Ohayon (2000a; 2000b)
Augmenting Thermodynamic Fields van Nostrand and Inman (1995)
Anelastic Displacement Fields Lesieutre and Lee (1996), Trindade, Benjeddou, and
Ohayon (2000a; 2000b; 2001a)
with the FE model equations of motion. While ATF/ADF models lead directly to time-
domain first order differential equations, it is in such a way that compatibility between these
equations and FE model ones is only obtained by building a coupled system in state space
form. As shown by Trindade, Benjeddou, and Ohayon (2000b), even if GHM and ADF
models use different parameters having different physical meanings, their resulting coupled
models present very similar responses.
Other methods also based on the inclusion of internal variables can be found in the
literature (Johnson, Tessler, and Dambach, 1997; Yiu, 1993). Table 3 presents a summary
of the viscoelastic damping models used in the literature of hybrid damping. Generally, these
models represent well the variation of viscoelastic materials properties with frequency for a
given constant temperature. To allow the representation of the temperature dependence of the
properties and the self-heating of viscoelastic materials, Lesieutre and his group (Brackbill
et al., 1996; Lesieutre and Govindswamy, 1996) extended the ADF model for these cases,
leading however, to nonlinear differential equations. Also, Baz (1998), Baz and Ro (1994),
Friswell and Inman (1998) and Trindade, Benjeddou, and Ohayon (2000a) have studied the
effects of operating temperature on hybrid damping treatments performance.
Since GHM and ADF models represent the viscoelastic material modulus by series of
functions in the Laplace-domain G(s) or frequency-domainG ∗(ω), it is of great importance
to determine accurately the parameters of the models from the material data. These
parameters, in general, are adjusted by a curve-fitting of the viscoelastic material master
curves, in order to minimize the difference between the measured and estimated data.
Lesieutre and Bianchini (1995) presented the curve-fitting of the ISD112 material data, at
a temperature of 27 C, in the frequency-range 8–8000 Hz. They concluded that five ADFs
(with two parameters per ADF) represent exactly the behavior of the material shear modulus
and the loss factor. Friswell, Inman, and Lam (1997) presented the same analysis for the
model of Golla and Hughes (1985), with three or four parameters per model. They used the
ISD112 material at 20 C in the frequency-range 10–4800 Hz and Dyad601 material at 24 C
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 711
in the frequency-range 2–4800Hz. Their results indicated that themodelwith four parameters
represented the material data better than that with three parameters, although it introduced
non-symmetrical matrices into the global model. In general, ADF and GHM models fit
well the master curves of materials whose properties have strong frequency dependence.
Nevertheless, the number of parameters needed is almost inversely proportional to the degree
of frequency dependence of the material properties. That is why Enelund and Lesieutre
(1999) proposed a combination of the ADF model with the fractional derivatives model in
order to represent the frequency dependence of weak material properties as well.
The main disadvantage of ADF and GHM models is that, associated with a FE
discretization, they lead to large systems, since they add auxiliary dof to account for the
frequency dependence of the viscoelastic material. Consequently, it is generally necessary to
reduce the dimension of the model by projection in a suitable reduced modal base. In order
to eliminate the relaxation modes, Park, Inman, and Lam (1999) used a method, named the
Modified Internal Balancing Method, combining the Internal Balancing Method, well-known
in the control area, with Guyan’s method to eliminate the less controllable and observable
modes, on one hand, and to rewrite the reduced model in terms of a subspace of FE nodal
variables, on the other hand. Friswell and Inman (1999) have presented a comparative
analysis of reduction methods by eigensystem truncation and balanced realization of GHM-
based models.
Comparisons between MSE, ADF and GHM viscoelastic models have been performed
for given (Trindade, Benjeddou, and Ohayon, 2000b) and variable (Trindade, Benjeddou,
and Ohayon, 2000a) operating temperatures. It was found that both ADF and GHM
models are superior to MSE model for highly damped structures and time-domain control
design. Moreover, ADF leads to smaller systems compared to GHM for similar frequency-
dependence representation performance. That is why the ADF model, associated with a
complex reduction modal base, retaining the first elastic modes of the structure (Trindade,
Benjeddou, and Ohayon, 2000b), will be used here.
OKQK ^ÅíáîÉ `çåíêçä ^äÖçêáíÜãë
For control systems design, a mathematical model of the structure must generally be
constructed and the required performances and disturbances to which the structure will be
subjected should be defined. The control design is very dependent on the choice and the
relative positions of actuators and sensors. It is well-known that, for collocated actuators
and sensors, even simple systems like proportional or derivative feedbacks lead to stable
systems (Preumont, 1997). For this reason, self-sensing actuators (Dosch, Inman, andGarcia,
1992) are interesting for active vibration control although they are not much used. For non-
collocated systems, more complex control systems, such as optimal control, must be used.
The control performance is also strongly dependent on the control algorithm used to
process the data provided by the sensors in order to determine the control signals sent to the
actuators. Several control algorithms have been used in the literature of hybrid active-passive
damping treatments. The proportional (P), derivative (D) and proportional-derivative (PD)
feedback algorithms are the most used ones, due to their simple design and implementation.
Kapadia andKawiecki (1997) considered a control proportional to the beam tip deflection, for
the ACL configuration of Figure 1(f). Azvine, Tomlinson, and Wynne (1995), Badre-Alam,
Wang, and Gandhi (1999) and Rongong et al. (1997) used a Direct Velocity Feedback (DVF)
712 M. A. TRINDADE and A. BENJEDDOU
of the beam tip deflection. However, the preceding control systems are non-collocated since,
in general, the piezoelectric actuator is located close to the clamped end, while the deflection
measurement is taken by means of an accelerometer or an optical sensor placed at the beam
tip. Consequently, the modes which are not in phase with the couple actuator/sensor may be
destabilized, as highlighted in theseworks. Rongong et al. (1997) observed that the additional
passive damping, provided by the viscoelastic layer, guaranteed a larger stability margin for
the modes excited by the controller, thus avoiding their destabilization. Badre-Alam, Wang,
and Gandhi (1999) proposed to filter the tip velocity before sending it to the controller, in
order to eliminate its periodic contribution due to the out-of-plane rotation of the beam. On
the other hand, some authors have used the derivative feedback of a piezoelectric sensor
voltage. Hence, Huang, Inman, and Austin (1996) and Yellin and Shen (1996) considered
a self-sensing actuator in the ACL treatment, where the derivative of the voltage induced in
the actuator is fed back after amplification as a control voltage. In these cases, the system
is perfectly collocated and, therefore, stable for any amplification. Indeed, Yellin and Shen
(1996) showed that this combination guarantees a positive energy dissipation of the system.
Other works have also presented a combination of the preceding algorithms, using
a proportional-derivative feedback of a displacement or sensor voltage. Shen (1994)
considered a PD algorithm of a cantilever beam tip deflection. He showed that, for the
first eigenmode, this control leads to better damping performances, compared to the simply
proportional one, but also to a lower ratio between hybrid and active treatment performances.
Baz and Ro (1995b) made use of the voltage and its derivative, provided by the piezoelectric
sensor bonded between the viscoelastic layer and the beam, to supply the control voltage to
the piezoelectric actuator of the ACL treatment. Then, in a later work (Baz and Ro, 1995a),
they used an optimization algorithm to evaluate the optimal proportional and derivative gains
that minimize a quadratic performance index of the system. This was also presented later by
Veley and Rao (1996). Varadan, Lim, and Varadan (1996) also considered a PD algorithm
but for a non-collocated system, since their piezoelectric sensor was placed beside the ACL
treatment (Figure 1(b)). A discussion on the destabilization of the modes excited by a PD
controller was presented by Lesieutre and Lee (1996). They proposed segmenting the ACL
treatment to obtain amore robust control system, so that the number of treatments corresponds
to the minimal quantity of modes damped by the controller.
Optimal algorithms, such as the Linear Quadratic Regulator (LQR), applied to hybrid
treatments were also used in the literature, although less often. Lam, Inman, and Saunders
(1997), Liao and Wang (1996; 1997a; 1997b) and Trindade, Benjeddou, and Ohayon (2000a;
2000b;2001a) adopted LQR control for ACL-treated beams, although in Lam, Inman, and
Saunders (1997), and Trindade, Benjeddou, and Ohayon (2000a; 2000b; 2001a), the cost
functionwas built in terms of the state variables and, in the others, it was in terms of the output
vector. Trindade, Benjeddou, and Ohayon (2000a) proposed a temperature dedicated LQR
control for an ACL-treated cantilever beam leading to uniform damping performances over
an operating temperature range. Tsai and Wang (1997) also used a LQR optimal algorithm
to study an ACL treatment whose piezoelectric actuator was connected to a shunting circuit
to increase the energy dissipation. In their case, as each group of parameters of the shunting
circuit provided a different optimal gain, a sequential quadratic programmingwas used to find
the parameters minimizing further the cost function. However, in all the preceding cases, the
voltage applied to the actuator is proportional to the state variables, which, consequently, must
be measurable. This assumption is rather constraining and not very realistic, although this
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 713
problem could be solved by taking into account a state observer, leading to a Linear Quadratic
Gaussian (LQG) type algorithm. Nevertheless, LQG controllers had not yet been applied
to hybrid damping treatments, but have been recently by the present authors (Trindade,
Benjeddou, and Ohayon, 2001b). In the present study, a LQR controller, accounting for
piezoelectric actuator control voltage limitations, will be used through an iterative algorithm
presented in Trindade, Benjeddou, and Ohayon (1999; 2000a).
Friswell and Inman (1998) employed the Positive Position Feedback (PPF) control
algorithm to reduce the vibrations of a beam with a PCL treatment, bonded on its upper
surface, and a piezoelectric self-sensing actuator, bonded on the opposite surface. This
algorithm allows each mode to be controlled separately, although, in their work, Friswell
and Inman (1998) considered only the first one. van Nostrand, Knowles, and Inman (1993)
also applied a PPF-like control algorithm to ACL-treated beams. The main advantage of
this algorithm is that its stability depends only on a knowledge of the open-loop system
eigenfrequencies.
Baz and his co-workers have been much interested in the use of algorithms adapted to
the specific case of hybrid treatments. Thus, Baz (1998) adopted a robust control algorithm
in order to guarantee the stability of the system, when subjected to uncertainties on the
viscoelastic material properties, due to variations in operating temperature, while minimizing
the H2 norm of the transfer function between the perturbation force and the beam tip
deflection. This has led to a second order controller with a uniform performance for the three
temperatures studied. Crassidis, Baz, and Wereley (2000) considered also a robust control,
but using the H∞ norm. In addition, in order to build an algorithm able of guaranteeing
energy dissipation, Baz (1997a) and Shen (1997) proposed a Boundary Controller by feedback
of the amplified relative axial velocity of the actuator edges. This velocity was supposed
measurable, which can only be done through a self-sensing actuator, or, at least, from relation
with the beam strain measurement. This algorithm was also adopted by Liu andWang (1998)
and then generalized by Baz (1997b) by multiplying the relative velocity of the edges by
arbitrary transfer functions, instead of the preceding constant gains, although only second and
third order functions were considered in Baz (1997b). This controller, named the Dynamic
Boundary Controller, provided better performances than those of its static correspondent. A
summary of the control algorithms used in the literature of hybrid damping is given in Table 4.
3. PARAMETRIC ANALYSIS OF HYBRID ACTIVE-PASSIVE DAMPING
TREATMENTS
This section aims at providing a numerical analysis of the damping performances of ACL,
APCL, AC/PCL and AC/PSOL treatments. Thus, a geometric optimization of the hybrid
treatments is studied, through parametric analyses, and a comparison of both their optimal and
overall damping performances is carried out. For that, the unified sandwich/multilayer beam
FE model, presented in Trindade, Benjeddou, and Ohayon (2001a), together with the ADF
viscoelastic model, are used to build state space models for beams partially covered by ACL,
APCL, AC/PCL and AC/PSOL hybrid damping treatments. Then, the resulting complex-
based modal reduced model (Trindade, Benjeddou, and Ohayon, 2000b) is applied to the
control design, using the iterative LQR control algorithm proposed in Trindade, Benjeddou,
and Ohayon (2000a).
714 M. A. TRINDADE and A. BENJEDDOU
Table 4. Control algorithms used in the literature of hybrid damping.
Proportional and/or Derivative Baz and Ro (1995a; 1995b), Huang, Inman, and Austin
(1996), Lesieutre and Lee (1996), Shen (1994),
Trindade, Benjeddou, and Ohayon (2001b), Varadan,
Lim, and Varadan (1996), Veley and Rao (1996),
Yellin and Shen (1996)
Direct Velocity Feedback Azvine, Tomlinson, and Wynne (1995), Badre-Alam,
Wang, and Gandhi (1999), Kapadia and Kawiecki (1997),
Rongong et al. (1997), Trindade, Benjeddou, and Ohayon
(2001b)
Linear Quadratic Regulator Lam, Inman, and Saunders (1997), Liao and Wang (1996;
1997a; 1997b), Trindade, Benjeddou, and Ohayon (2000a;
2000b; 2001a), Tsai and Wang (1997)
Linear Quadratic Gaussian Trindade, Benjeddou, and Ohayon (2001b)
Positive Position Feedback Friswell and Inman (1998), van Nostrand, Knowles, and
Inman (1993)
Robust control Baz (1998), Crassidis, Baz, and Wereley (2000)
Boundary control Baz (1997b), Liu and Wang (1998), Shen (1997)
PKNK mêçÄäÉã aÉëÅêáéíáçå ~åÇ ^å~äóëáë jÉíÜçÇçäçÖó
A cantilever aluminum beam with length L = 280 mm, width b = 25 mm and thickness
hb = 3mm is considered. It is subjected to an impulsive transverse perturbation force applied
at the beam tip, which magnitude is set to induce a maximum tip deflection amplitude of
1.5 mm. In addition, the beam tip deflection is measured through a point sensor for analysis
purposes. However, for control design purposes, all variables are supposed to be measured
since a full state feedback control algorithm LQR is considered. Clearly, this would require an
excessive number of sensors in practice, but a state observer may be coupled to the controller
without loss of performance (Trindade, Benjeddou, and Ohayon, 2001b). Using four hybrid
active-passive damping treatments, namely ACL, APCL, AC/PCL and AC/PSOL, combined
with an iterative LQR control algorithm, the first three bending eigenmodes are to be damped.
This is done using a PZT5H piezoelectric actuator, an ISD112 viscoelastic layer and, except
for the ACL treatment, an aluminum constraining layer. A viscous damping of 0.1% is
considered to represent all sources of damping other than the viscoelastic one. The reduced-
order model is constructed using the first five damped bending eigenmodes. First, the
optimization of each treatment, through a parametric analysis, is presented. The treatment
length and the viscoelastic layer thickness are varied in the ranges 20–70mm and 0.01–2mm,
respectively. The former is limited to 70 mm because it corresponds to the maximum length
of commercially available piezoceramic actuators. Details on the length optimization of the
passive constrained layer dampingmay be found in Plunkett and Lee (1970). The viscoelastic
layer thickness is limited to 2 mm in order to save material and to limit the total weight of
the structure. A 10 mm distance is considered between the left edge of the treatment and the
clamped end of the beam. The optimization is performed to maximize the sum of the first
three passive and hybrid modal damping factors.
The piezoelectric actuator thickness is set to hp = 0.5 mm and its maximum control
voltage to 250 V, thus leading to a maximum electrical field of 500 V/mm. The plane-
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 715
hv
hp
a
cdL
hbAluminum
PZT5HISD112
Figure 3. Cantilever beam partially covered with active constrained layer damping treatment.
stress modified properties of aluminum are: E ′
b = 79.8 GPa and ρ = 2690 kg m−3 and
those of the piezoelectric material PZT5H are: E ′
p = 65.5 GPa, ρ = 7500 kg m−3 and
e ′
31= −23.2 C m−2. The viscoelastic material properties are represented by a three-series
ADF model with the following parameters: G0 = 0.50MPa, ∆ = [0.746; 3.265; 43.284]and Ω = [468.7; 4742.4; 71532.5] rad/s, and its mass density is ρ = 1600 kg m−3. The
operation temperature is supposed to be uniform, constant and equal to 27 C. In order to
damp the first three modes, the LQR state weight matrix is set toQ = diag(1, 1, 1, 0, . . . , 0)and the input weight is evaluated by the iterative LQR algorithm to respect the above voltage
limitation.
PKOK ^ÅíáîÉ `çåëíê~áåÉÇ i~óÉê a~ãéáåÖ
The ACL treatment is obtained by bonding an ISD112 layer on the upper surface of the beam,
then a PZT5H actuator on the viscoelastic layer, as shown in Figure 3.
Passive (open-loop) and hybrid (closed-loop) modal damping factors provided by the
ACL treatment are studied for various treatment lengths and viscoelastic layer thicknesses,
in order to obtain an optimal construction. Figure 4 presents the variation of the sum of the
first three modal damping factors in open-loop ζp with the treatment length a and viscoelasticlayer thickness hv . It shows that passive damping is optimal for long treatments (a = 70mm)
and very thin viscoelastic layer (hv = 0.03 mm). Nevertheless, it is worthwhile to note that
the optimal thickness grows with the treatment length. Also, one may notice from Figure 4
that the passive damping dependence on viscoelastic thickness is more important than that
on treatment length. That is why the damping performance is adequate in the whole length
range for thin viscoelastic layers.
Next, the piezoelectric actuator of the ACL treatment is activated through connection
with the LQR controller to enhance the passive performance. In an analysis similar to that for
the open-loop system, the influence of the treatment length a and viscoelastic layer thicknesshv on the sum of the first three closed-loopmodal damping factors ζh is presented in Figure 5.It is observed that hybrid damping is also optimal for long treatments (a = 70mm) and very
thin viscoelastic layers (hv = 0.02 mm). Figure 5 also shows that hybrid damping, just
like passive one (Figure 4), is effective only for very thin viscoelastic layers. This is due
to the poor passive damping performance for thick viscoelastic layers, and the controller
716 M. A. TRINDADE and A. BENJEDDOU
10−2
10−1
100
20
40
60
0
2
4
6
8
hv (mm) a (mm)
ζ p (%
)
Figure 4. Influence of the treatment length a and viscoelastic layer thickness hv on the sum of the first
three open-loop modal damping factors ζp of the ACL-treated beam.
10−2
10−1
100
20
40
60
0
5
10
15
hv (mm) a (mm)
ζ h (%
)
Figure 5. Influence of the treatment length a and viscoelastic layer thickness hv on the sum of the first
three closed-loop modal damping factors ζh of the ACL-treated beam.
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 717
10−2
10−1
100
20
40
60
0
50
100
150
hv (mm) a (mm)
ζ a (%
)
Figure 6. Influence of the treatment length a and viscoelastic layer thickness hv on the damping gain
ζa for the three first modes (relative to open-loop) of the ACL-treated beam.
performance decrease with the thickness of the viscoelastic layer. Figure 6 presents the
damping gain (or the contribution of active damping) provided by the controller, evaluated
by ζa = ζh/ζp −1. It confirms the loss of effectiveness of the piezoelectric actuator with the
viscoelastic layer thickness increase. This is due to the fact that the transmissibility between
the actuator and the beam is reduced by the softness of the viscoelastic layer. Indeed, Figure 7
shows that the controllability of both first and second modes decreases strongly with the
viscoelastic layer thickness. It is thus clear that this treatment is advantageous compared to
the PCL only when the latter is very limited in thickness. Hence, a good active performance
is obtained only in spite of the passive one, leading to less robust systems. Nevertheless, in
order to guarantee a good performance in open-loop, the sum of passive ζp and hybrid ζhdamping factors are used as measurement of optimality, although the latter index (ζp + ζh )double counts the passive damping effects. Thus, according to Figures 4 and 5, the optimal
ACL configuration is obtained for a = 70 mm and hv = 0.03 mm, although in this case
the active controller does not increase too much the damping performance compared to the
open-loop one.
Next, an analysis of the attainable performances of the optimal ACL treatment is
performed. Hence, two cases are studied in detail, where the LQR controller is designed
to optimize the damping of the first mode, then that of the second one. Figure 8 shows the
frequency response of the beam untreated and treated by an ACL in open-loop (acting like a
PCL) and in closed-loop. The iterative LQR control algorithm is designed to optimize the first
mode bymakingQ = diag(1, 0, . . . , 0) and, the second one, withQ = diag(0, 1, 0, . . . , 0).One observes, in Figure 8, that it is possible to increase the damping of the chosen mode
without losing the passive damping of the other modes of the beam. Let us note that, even
purely passive damping, provided by the inactive ACL, reduces reasonably the amplitude
718 M. A. TRINDADE and A. BENJEDDOU
10−2
100
2040
600
0.5
1
hv (mm) a (mm)
B(1
)
10−2
100
2040
600
0.5
1
hv (mm) a (mm)
B(2
)Figure 7. Influence of the treatment length a and viscoelastic layer thickness hv on the controllability
of the first B(1) and second B(2) modes of the ACL-treated beam (B(i) is the ith component of the
reduced control vector).
102
103
−200
−180
−160
−140
−120
−100
−80
−60
−40
Am
plitu
de (
dB)
Frequency (Hz)
Base beam ACL inactiveACL mode 1ACL mode 2
Figure 8. Open- and closed-loop FRF of the ACL-treated beam with optimal configuration (a = 70 mm,
hv = 0.03 mm).
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 719
0 0.1 0.2 0.3 0.4 0.5−1.5
−0.75
0
0.75
1.5
Tip
def
lect
ion
(mm
)
Time (sec)
ACL inactiveACL mode 1
Figure 9. Open- and closed-loop transient responses of the ACL-treated beam with optimal configuration
(a = 70 mm, hv = 0.03 mm).
of resonances compared to the untreated beam. The first and second modes are damped by
their controllers by 9 dB (4.7%) and 10 dB (4.8%), respectively. The controllers of the first
and second modes excite the second and the fourth anti-resonances but without exciting any
resonance corresponding to the other modes.
The transient response of the beam tip deflection is also presented for the optimal case
in Figure 9. It shows that the hybrid controller (optimized to control the first mode) more
than doubles the attenuation speed of the transverse vibration of the beam (0.35 seconds for
the active ACL against 0.80 seconds for the inactive ACL). The controller optimized for the
second mode is not represented in this figure because its response is almost identical to that
of the open-loop case. This may be explained by the fact that the second mode contributes
very little to the beam tip deflection and, therefore, its control does not help to attenuate the
beam deflection. The control voltage in the piezoelectric actuator for the two controllers is
presented in Figure 10. One may notice that the two control voltages are well limited to the
maximum voltage of the piezoelectric actuator (250 V). However, for the controller of the
first mode, the voltage is damped as fast as the beam closed-loop response (Figure 9), while
the control voltage of the second mode is canceled much more quickly than the open-loop
response of the beam. Indeed, this time corresponds to the time necessary for this controller
to eliminate the second mode vibration. Nevertheless, it is not observable in the closed-loop
response since the latter is almost only made of the first mode contribution
It is also worthwhile to study, for the optimal ACL configuration, the influence of
treatment segmentation on modal damping. For that, one supposes that the treatment length,
a = 70 mm, can be divided into up to five segments depending on the same controller but
with individual voltages. Nevertheless, in order to respect the FEmodel assumptions, a 5 mm
spacing is considered between the segments. Figure 11 presents the variation of the first three
passive and hybrid modal damping factors with the number of treatment segments. It shows
that treatment segmentation does not improve passive damping of the first three modes, in
particular for 2 and 3 segments where damping is rather reduced. Moreover, additional active
damping is more effective for one segment treatment. The reason for this is that the limiting
720 M. A. TRINDADE and A. BENJEDDOU
0 0.1 0.2 0.3 0.4 0.5−250
−125
0
125
250C
ontr
ol v
olta
ge (
V)
Time (sec)
ACL mode 1ACL mode 2
Figure 10. Control voltages to enhance damping of first and second modes of the ACL-treated beam
with optimal configuration (a = 70 mm, hv = 0.03 mm).
1 2 3 4 52
3
4
5
6
7
Number of segments
Mod
al d
ampi
ng (
%) Mode 1
Mode 2Mode 3
Figure 11. Variation of passive (dashed line) and hybrid (solid line) modal damping factors with treatment
segmentation (a = 70 mm and hv = 0.03 mm).
voltage is reached only in one of the actuators, which is the optimal one (that near the clamped
end for this example) to control the large weighted modes. Therefore, the other actuators
are not fully used, such that maximum voltages are: V1 = 251.02 V, V2 = 214.77 V,
V3 = 162.99 V, V4 = 117.35 V and V5 = 47.50 V. The controllability of the first three
modes by the first actuator (that closest to the clamped end) for various segmentation is
presented in Figure 12. It may be observed that this actuator is optimal when it is alone.
Consequently, one can conclude that the segmentation associated with this control algorithm
is not effective due to the reduction of fully used active surface. Nevertheless, this conclusion
may not be valid for other control strategies (Lesieutre and Lee, 1996).
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 721
1 2 3 4 50.2
0.4
0.6
0.8
1
Number of segments
Fir
st a
ctua
tor
cont
roll
abil
ity
B(1) B(2) B(3)
Figure 12. Variation of the first three modes controllability by the first actuator with treatment segmentation
(a = 70 mm et hv = 0.03 mm) (B(i) is the ith component of the reduced control vector).
a
cdL
hb
hv
chhp
Aluminum
ISD112AluminumPZT5H
Figure 13. Cantilever beam partially covered with active-passive constrained layer damping treatment.
According to these results, one can conclude that the main disadvantage of the ACL
treatment is that the performance of hybrid control, compared to that of the passive, can be
increased only by reducing much the thickness of the viscoelastic layer. Consequently, it is
useful when the passive damping of the first modes is less important than the hybrid one, or
when technical or economic constraints limit the use of passive material at least.
PKPK ^ÅíáîÉJm~ëëáîÉ `çåëíê~áåÉÇ i~óÉê a~ãéáåÖ
Sometimes, it is interesting to bond the piezoelectric actuator on an already existing PCL
treatment that is applied on a surface of the structure (Figure 13). In this case, the actuator
augments the constraining layer instead of replacing it as in the previous treatment, leading
to an APCL treatment. Although this treatment consists generally in different lengths for the
active and passive layers, for comparison purposes, the same length awill be considered herefor both PCL and actuator layers. Here, the material and geometrical data are augmented by
722 M. A. TRINDADE and A. BENJEDDOU
10−2
10−1
100
20
40
60
0
2
4
6
8
10
hv (mm) a (mm)
ζ p (%
)
Figure 14. Influence of the treatment length a and viscoelastic layer thickness hv on the sum of the
first three open-loop modal damping factors ζp of the APCL-treated beam.
the thickness of the aluminum constraining layer hc = 0.5 mm. As in the preceding sub-
section, hybrid damping performances of this treatment are now analyzed.
Open- and closed-loop modal damping of the APCL-treated beam are evaluated for
several treatment lengths a and viscoelastic layer thicknesses hv. Figure 14 presents the
influence of these two parameters on the sum of the first three passive (open-loop) modal
damping factors ζp . It shows that the damping variation is similar to that of theACL treatment
(Figure 4), although the present treatment (APCL) leads to higher damping factors. This can
be explained by the fact that, for the inactive piezoelectric actuator, the effective constraining
layer is composed by the piezoelectric and elastic layers. Therefore, since these layers have
the same thickness, the resulting constraining layer is twice as thick as that of inactive ACL,
leading to more effective passive damping. Figure 14 shows also that the optimal passive
performance is obtained for very thin viscoelastic layers (hv = 0.03mm) and long treatments
(a = 70 mm). As for the preceding case, the optimal thickness increases with the treatment
length.
In order to enhance the passivemodal damping, the LQR controller is used in conjunction
with the piezoelectric actuator. As shown in Figure 15, modal damping is well improved
by the controller compared to the open-loop case. This figure shows also that the sum
of the first three hybrid (closed-loop) modal damping factors ζh is optimal for very thin
viscoelastic layers (hv = 0.03 mm) and long treatments (a = 70 mm), just as for the
passive one. However, the hybrid damping performance is less dependent on the viscoelastic
layer thickness than the passive one. Also, on the opposite of the preceding case, the
active damping gain (ζa = ζh/ζp − 1) provided by the piezoelectric actuator increases
for thick viscoelastic layers and short treatments, as shown in Figure 16. This is mainly
because passive damping is less effective in that case, while hybrid damping performance
is more uniform in the thickness and length ranges. One may also notice from Figure 16
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 723
10−2
10−1
100
20
40
60
0
5
10
15
20
hv (mm) a (mm)
ζ h (%
)
Figurq 15. Influence of the treatment length a and viscoelastic layer thickness hv on the sum of the
first three closed-loop modal damping factors ζh of the APCL-treated beam.
10−2
10−1
100
20
40
60
0
100
200
300
400
500
hv (mm) a (mm)
ζ a (%
)
Figure 16. Influence of the treatment length a and viscoelastic layer thickness hv on the damping gain ζa
for the three first modes (relative to open-loop) of the APCL-treated beam.
724 M. A. TRINDADE and A. BENJEDDOU
10−2
100
2040
600
0.5
1
hv (mm) a (mm)
B(1
)
10−2
100
2040
600
0.5
1
hv (mm) a (mm)
B(2
)
Figure 17. Influence of the treatment length a and viscoelastic layer thickness hv on the controllability
of the first B(1) and second B(2) modes of the APCL-treated beam (B(i) is the ith component of the
reduced control vector).
that the passive modal damping can be enhanced up to 400% by the controller, although
this maximum improvement is obtained in the worst case. In fact, the optimal performance,
obtained by the maximization of the sum of passive and active damping factors, is achieved
for very thin viscoelastic layers (hv = 0.03 mm) and long treatments (a = 70 mm). In this
case, modal damping is improved actively by only 88%. It is worthwhile noticing also that,
for long treatments, the hybrid (closed-loop) damping is less dependent on the viscoelastic
layer thickness compared to the ACL treatment. This is due to the smaller controllability
loss of the APCL actuator for thick viscoelastic layers, as shown in Figure 17. Indeed, the
controllability of the first mode increases strongly with the treatment length, as for the ACL
treatment, but it does not decrease as much as that of ACL for increasing viscoelastic layer
thickness, so that it is ten times higher than that of ACL for the extreme case of hv = 2 mm
and a = 70 mm.
The frequency-domain impulsive responses of the optimal APCL-treated beam (hv =0.03 mm and a = 70 mm) in open- and closed-loop are evaluated. The frequency response
function of the tip deflection of the beam loaded by the perturbation transverse force is
shown in Figure 18. Four cases are shown, that of the base beam (without passive or active
treatments), that of the beam treated by the APCL in open-loop and those of the APCL-
treated beam with controllers optimized to control the first and the second modes. The LQR
controller is optimized for the first or second modes by setting the weight matrix Q as
presented in the previous sub-section. The active control allows us to decrease the amplitude
of the selected mode resonance. The first and second modes amplitudes are attenuated by
10.4 dB (7.74%) and 10 dB (6.20%), respectively. It can also be observed in Figure 18 that
both open- and closed-loop responses are more damped than those corresponding to the ACL
treatment.
The open- and closed-loop transient responses of the APCL-treated beam with optimal
configuration (a = 70 mm, hv = 0.03 mm) are shown in Figure 19. As for the preceding
case, only the closed-loop response using the first mode controller is shown, since the second
mode controller does not affect the output response. It can be noticed that the first active
controller allows a decrease of the beam tip response settling time (instant of time at which
the output converges to 2% of its maximum value) to 30% of that in open-loop (0.20 seconds
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 725
102
103
−200
−180
−160
−140
−120
−100
−80
−60
−40
Am
plitu
de (
dB)
Frequency (Hz)
Base beam AC/PSOL inactiveAC/PSOL mode 1 AC/PSOL mode 2
Figure 18. Open- and closed-loop FRF of the APCL-treated beam with optimal configuration (a = 70mm,
hv = 0.03 mm).
0 0.1 0.2 0.3 0.4 0.5−1.5
−0.75
0
0.75
1.5
Tip
def
lect
ion
(mm
)
Time (sec)
AC/PSOL inactiveAC/PSOL mode 1
Figure 19. Open- and closed-loop transient responses of the APCL-treated beam with optimal
configuration (a = 70 mm, hv = 0.03 mm).
in closed-loop against 0.70 seconds in open-loop). In fact, even the open-loop response of
APCL is faster than that of ACL, since the passive damping of the former is more effective.
However, while the open-loop response settling time is reduced by only 13%, compared to
the ACL treatment, the closed-loop one is 43% smaller than the ACL corresponding one.
The first and second controller voltages for the APCL-treated beam are shown in Figure 20.
One may observe that both controller voltages are limited to 250 V by the iterative LQR
algorithm. Also, as for the preceding case and for the same reason, the second controller
voltage vanishes faster than the first controller one.
726 M. A. TRINDADE and A. BENJEDDOU
0 0.1 0.2 0.3 0.4 0.5−250
−125
0
125
250C
ontr
ol v
olta
ge (
V)
Time (sec)
APCL mode 1APCL mode 2
Figure 20. Control voltages to enhance damping of first and second modes of the APCL-treated beam
with optimal configuration (a = 70 mm, hv = 0.03 mm).
It was shown in this sub-section that the APCL treatment leads to better performances
than the ACL one in both open- and closed-loop. This is mainly due to the increase
of constraining layer thickness and the reduction of the dependence of the actuator
controllability on the viscoelastic layer thickness. However, APCL treatments are thicker
than the corresponding ACL ones, leading to total weight, thickness and cost increase. Using
elastic constraining layers, other treatments are proposed in the next two sub-sections by
varying the relative position of the PCL treatment and the piezoelectric actuator.
PKQK ^ÅíáîÉ `çåíêçä ~åÇ m~ëëáîÉ `çåëíê~áåÉÇ i~óÉê a~ãéáåÖ
In this sub-section, one considers an alternative to the preceding configurations where the
active and passive treatments do not act directly together, but separately. This consists of a
PCL treatment bonded on a surface of the beam and a piezoelectric actuator bonded at the
same position but on the opposite surface (Figure 21). The damping performance of this
hybrid treatment is presented here, using similar analyses to those performed in the two last
sub-sections.
A parametric analysis of the first three open- and closed-loop modal damping factors of
the cantilever beam of Figure 21 is now carried out. Just like for the preceding treatments,
passive (open-loop) damping performance is optimal for long treatments (a = 70 mm) and
very thin viscoelastic layer (hv = 0.03 mm), as shown in Figure 22. However, in open-
loop, this treatment is less effective than inactive ACL and APCL. This can be due to the
presence of the piezoelectric actuator bonded on the lower surface of the beam that stiffens
the structure without increasing its damping. This is why the damping performances of the
ACL- and APCL-treated beams are, respectively, up to 24% and 56% superior to that of the
present AC/PCL-treated one for short treatments (a = 20 mm) and thin viscoelastic layers
(hv = 0.1 mm).
Although the variation with the treatment length and viscoelastic layer thickness of
the passive damping of beams treated by ACL and AC/PCL is similar, the closed-loop
performances of these treatments are very different. Figure 23 shows that the hybrid
damping of AC/PCL treatment is optimal for long treatments (a = 70 mm) and almost
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 727
hv
hb
cdhp
hc
ISD112Aluminum
PZT5H
a
L
Aluminum
Figure 21. Cantilever beam partially covered with active control and passive constrained layer damping
treatment.
10−2
10−1
100
20
40
60
0
2
4
6
8
hv (mm) a (mm)
ζ p (%
)
Figure 22. Influence of the treatment length a and viscoelastic layer thickness hv on the sum of the
first three open-loop modal damping factors ζp of the AC/PCL-treated beam.
independent on the viscoelastic layer thickness, although slightly more effective for very
thin viscoelastic layers (hv = 0.03 mm). Indeed, bonding the piezoelectric actuator directly
on the beam allows us to increase the transmissibility between them. Thus, the active control
authority is not reduced by the softness of the viscoelastic layer. This is also why the active
damping gain (ζa = ζh/ζp − 1) provided by the controller increases with the viscoelastic
layer thickness (Figure 24), in contrast to the ACL. According to Figure 24, the active
controller may enhance the modal damping up to 2300% for thick viscoelastic layer and
short treatments. Nevertheless, this improvement is mainly due to the bad performance of
the passive treatment in that case. In fact, the optimal global performance (ζp+ζh ) is obtainedfor very thin viscoelastic layers (hv = 0.03 mm) and long treatments (a = 70 mm), where
728 M. A. TRINDADE and A. BENJEDDOU
10−2
10−1
100
20
40
60
5
10
15
20
25
30
hv (mm) a (mm)
ζ h (%
)
Figure 23. Influence of the treatment length a and viscoelastic layer thickness hv on the sum of the
first three closed-loop modal damping factors ζh of the AC/PCL-treated beam.
10−2
10−1
100
20
40
60
0
500
1000
1500
2000
2500
hv (mm) a (mm)
ζ a (%
)
Figure 24. Influence of the treatment length a and viscoelastic layer thickness hv on the damping gain ζa
for the three first modes (relative to open-loop) of the AC/PCL-treated beam.
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 729
10−2
100
2040
600
0.5
1
hv (mm) a (mm)
B(1
)
10−2
100
2040
600
0.5
1
hv (mm) a (mm)
B(2
)
Figure 25. Influence of the treatment length a and viscoelastic layer thickness hv on the controllability of
the first B(1) and second B(2) modes of the AC/PCL-treated beam (B(i) is the ith component of the
reduced control vector).
the active controller enhancement is 340%. As guessed previously, Figure 25 shows that
the controllability of the AC/PCL treatment is almost independent of the viscoelastic layer
thickness. For short treatments (a = 20 mm), the controllability of the first mode is nearly
constant and, for long treatments, a = 70 mm, the actuator preserves 83% of its maximum
controllability for the thicker viscoelastic layer considered (hv = 2 mm).
For the optimal configuration (a = 70mm and hv = 0.03mm), Figures 26 and 27 show
the impulsive responses of the AC/PCL-treated beam, in frequency- and time-domain. As in
the preceding sub-sections, two controllers are considered in order to optimize, in one case
the damping of the first mode and in the other that of the second one. Figure 26 indicates that
the first controller attenuates the first mode resonance by 20 dB (19.3%), whereas the second
attenuates the second mode by 15 dB (11.3%). It is worthwhile noticing that these hybrid
performances are largely superior to those obtained with the preceding treatments. However,
the open-loop frequency response is much less passively damped.
Figure 27, presenting the transient response of the beam with AC/PCL treatment, shows
that the controller optimized to control the first mode attenuates the transverse vibration of
the beam much better compared to the open-loop case. However, the controller of the second
mode leads to identical responses in open- and closed-loop, since this mode contributes little
to the output response. Hence, it is not represented in this figure. One may also notice from
Figure 27 that the settling time of the transient response of the beam is strongly reduced by
the first controller (0.9 seconds in open-loop against 0.1 seconds in closed-loop). The fast
oscillations of the closed-loop output response observed in this figure are due to the fact that
the contribution of the first mode to the output transient response is almost eliminated, making
the other modes more observable. In fact, Figure 26 shows that the difference between the
resonance amplitudes of the first and second modes passes from 30 dB, in open-loop, to
10 dB, in closed-loop. Figure 28 shows that the voltage of the second controller vanishes
more quickly than that of the first one but both voltages are limited to 250 V, as required by
the iterative control algorithm.
These results show that the use of passive and active treatments separately generally leads
to higher performances than those obtained by using the actuator bonded on the viscoelastic
layer (ACL) or on the constraining layer of a PCL (APCL). This treatment also allows a good
730 M. A. TRINDADE and A. BENJEDDOU
102
103
−200
−180
−160
−140
−120
−100
−80
−60
−40A
mpl
itude
(dB
)
Frequency (Hz)
Base beam AC/PCL inactiveAC/PCL mode 1AC/PCL mode 2
Figure 26. Open- and closed-loop FRF of the AC/PCL-treated beam with optimal configuration
(a = 70 mm, hv = 0.03 mm).
0 0.1 0.2 0.3 0.4 0.5−1.5
−0.75
0
0.75
1.5
Tip
def
lect
ion
(mm
)
Time (sec)
AC/PCL inactiveAC/PCL mode 1
Figure 27. Open- and closed-loop transient responses of the AC/PCL-treated beam with optimal
configuration (a = 70 mm, hv = 0.03 mm).
performance to be retained for sufficiently thick viscoelastic layers, thus extending the choice
of passive damping. Moreover, in contrast to the ACL, the lengths of the passive and active
treatments can be different, thus allowing the consideration of passive treatments longer than
the limit imposed by the piezoelectric actuators. However, as APCL, this treatment leads
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 731
0 0.1 0.2 0.3 0.4 0.5−250
−125
0
125
250C
ontr
ol v
olta
ge (
V)
Time (sec)
AC/PCL mode 1AC/PCL mode 2
Figure 28. Control voltages to enhance damping of first and second modes of the AC/PCL-treated beam
with optimal configuration (a = 70 mm, hv = 0.03 mm).
to higher total thicknesses compared to ACL, since an elastic constraining layer over the
viscoelastic material is necessary to provide good passive performances.
PKRK ^ÅíáîÉ `çåíêçä ~åÇ m~ëëáîÉ pí~åÇJlÑÑ i~óÉê a~ãéáåÖ
An alternative to the treatment presented previously (AC/PCL) consists in bonding a
piezoelectric actuator between the PCL treatment and the beam, as shown in Figure 29.
Since the actuator elevates the PCL treatment, this leads to a PSOL damping treatment in the
open-loop case. The parametric analysis of open- and closed-loop modal damping realized
in the previous sub-sections is also performed for the present hybrid damping configuration
(AC/PSOL, Figure 29).
The variation of the sum of the first three passive (open-loop) modal damping factors
with the treatment length a and viscoelastic layer thickness hv is presented in Figure 30.
It shows that passive damping is optimal for thin viscoelastic layers (hv = 0.06 mm) and
long treatments (a = 70 mm). This treatment, when it is passive, is more effective up
to 50% compared to the ACL, 20% compared to APCL and 87% compared to AC/PCL,
although thicker than the previous ones, in particular for short treatments (a = 20 mm)
and thin viscoelastic layers (hv = 0.1 mm). Nevertheless, for long treatments and very
thin viscoelastic layers, inactive APCL is 12% more effective than inactive AC/PSOL. Lam,
Inman, and Saunders (1998) have found also that this treatment is more advantageous to
passively control the bending vibrations of a beam than the ACL and AC/PCL treatments,
although they have considered very long treatments of about 270 mm. According to the
present results (Figures 4, 22 and 30), this advantage is more visible for short treatments.
Just like the AC/PCL, this treatment is always more effective than the ACL one, in
particular for short treatments and thick viscoelastic layers (Figures 5, 23 and 31). This is
explained by the smaller dependence of the performance of this treatment on the viscoelastic
material thickness (Figure 31), compared to the ACL one (Figure 5). Figure 31 also shows
that AC/PSOL is optimal for long treatments (a = 70 mm) and less thinner viscoelastic
layers (hv = 0.2 mm) than those optimal of the preceding treatments (Figures 5, 15 and 23).
732 M. A. TRINDADE and A. BENJEDDOU
hvhp
cha
cdL
hbAluminum
PZT5HISD112Aluminum
Figure 29. Cantilever beam partially covered with active control and passive stand-off layer damping
treatment.
10−2
10−1
100
20
40
60
0
2
4
6
8
10
hv (mm) a (mm)
ζ p (%
)
Figure 30. Influence of the treatment length a and viscoelastic layer thickness hv on the sum of the
first three open-loop modal damping factors ζp of the AC/PSOL-treated beam.
Moreover, according to the same figure, the performance of the active controller does not
decrease with an increase in the viscoelastic layer thickness, although the hybrid damping of
this treatment is more dependent on this parameter than the AC/PCL. Consequently, as for
the latter, the damping gain provided by the piezoelectric actuator of the AC/PSOL increases
for thick viscoelastic layers, as shown in Figure 32. This figure also shows that the active
controller may improve modal damping by up to 1600% for thick viscoelastic layer and short
treatments. However, for the optimal configuration, this enhancement reduces to 230%,
which is less than that of AC/PCL but more than those of ACL and APCL. The optimal
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 733
10−2
10−1
100
20
40
60
5
10
15
20
25
30
hv (mm) a (mm)
ζ h (%
)
Figure 31. Influence of the treatment length a and viscoelastic layer thickness hv on the sum of the
first three closed-loop modal damping factors ζh of the AC/PSOL-treated beam.
10−2
10−1
100
20
40
60
0
500
1000
1500
2000
hv (mm) a (mm)
ζ a (%
)
Figure 32. Influence of the treatment length a and viscoelastic layer thickness hv on the damping gain ζa
for the three first modes (relative to open-loop) of the AC/PSOL-treated beam.
734 M. A. TRINDADE and A. BENJEDDOU
10−2
100
2040
600
0.5
1
hv (mm) a (mm)
B(1
)
10−2
100
2040
600
0.5
1
hv (mm) a (mm)
B(2
)
Figure 33. Influence of the treatment length a and viscoelastic layer thickness hv on the controllability of
the first B(1) and second B(2) modes of the AC/PSOL-treated beam (B(i) is the ith component of the
reduced control vector).
length and thickness are determined by the maximization of the sum of passive ζp and hybridζh damping factors. This leads to a = 70 mm and hv = 0.1 mm. This thicker optimal
viscoelastic layer may be due to the fact that, in contrast to the preceding treatments, here
the controllability is improved by augmenting the viscoelastic layer thickness (Figure 33).
Indeed, AC/PSOL controllability of the first mode augments with treatment length and
viscoelastic layer thickness and that of the second mode augments with the second parameter
and is optimal for medium length treatments.
The impulsive responses of the beam in the frequency- and time-domain are presented
for the optimal case (a = 70 mm and hv = 0.1 mm) in Figures 34 and 35. The two
controllers described in the sub-section 3.2 are used here. Figure 34 shows that the controllers
relative to the first and second modes lead to attenuation of 17 dB (16.3%) and 9 dB (6.1%),
respectively, compared to the open-loop system. Comparing Figures 26 and 34, one observes
that this treatment is less effective than the AC/PCL for attenuating the second mode actively.
However, the passive damping of this mode is higher for the present AC/PSOL treatment.
Figure 35 presents the transient response of the AC/PSOL-treated beam. It shows that
the first mode controller improves greatly the settling time of the beam transverse vibration
(0.12 seconds in closed-loop against 0.90 seconds in open-loop). However, the second mode
controller does not enhance the response of the beam. Through comparison of Figures 27 and
35, one observes that, in contrast to the AC/PCL, the fast oscillations are not present in the
closed-loop output response of the AC/PSOL. This is explained by the difference between
the resonance amplitudes of the first mode, in open- and closed-loop, compared to the other
modes. Indeed, the difference between the amplitudes of the first and second modes for the
AC/PCL passes from 30 dB, in open-loop, to 10 dB, in closed-loop (Figure 26). For the
present treatment (AC/PSOL), this difference passes from 33 dB to 16 dB (Figure 34). This
is due to the fact that, in contrast to the AC/PCL, the AC/PSOL is less effective in controlling
the first mode actively, while more effective in controlling the second one passively. Fig-
ure 36, presenting the control voltage of the first and second modes controllers, shows that
the voltage of the latter is canceled less quickly than that of the preceding treatment AC/PCL
(Figure 28), but more quickly than that of the first mode. Also, both control voltages are
within the limit voltage of 250 V.
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 735
102
103
−200
−180
−160
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−120
−100
−80
−60
−40A
mpl
itude
(dB
)
Frequency (Hz)
Base beam AC/PSOL inactiveAC/PSOL mode 1 AC/PSOL mode 2
Figure 34. Open- and closed-loop FRF of the AC/PSOL-treated beam with optimal configuration
(a = 70 mm, hv = 0.1 mm).
0 0.1 0.2 0.3 0.4 0.5−1.5
−0.75
0
0.75
1.5
Tip
def
lect
ion
(mm
)
Time (sec)
AC/PSOL inactiveAC/PSOL mode 1
Figure 35. Open- and closed-loop transient responses of the AC/PSOL-treated beam with optimal
configuration (a = 70 mm, hv = 0.1 mm).
736 M. A. TRINDADE and A. BENJEDDOU
0 0.1 0.2 0.3 0.4 0.5−250
−125
0
125
250C
ontr
ol v
olta
ge (
V)
Time (sec)
AC/PSOL mode 1AC/PSOL mode 2
Figure 36. Control voltages to enhance damping of first and second modes of the AC/PSOL-treated
beam with optimal configuration (a = 70 mm, hv = 0.1 mm).
The present results have shown that, in this treatment, the actuator acts directly on the
elastic beam and modifies the behavior of the PCL treatment under which it is placed. The
AC/PSOL hybrid damping performance is better than those of ACL and APCL but worse
generally than that of AC/PCL. However, the optimal performance index, defined as the sum
of passive and hybrid damping factors, shows that AC/PSOL may be more advantageous
than AC/PCL. Detailed comparisons between the hybrid damping treatments analyzed in this
section are presented in the next one.
4. COMPARISONOFHYBRIDACTIVE-PASSIVEDAMPINGTREATMENTS
In this section, comparisons between the hybrid damping treatments ACL, APCL, AC/PCL
and AC/PSOL are presented. Hence, the optimal open- and closed-loop performances of the
hybrid treatments are compared; then, the differences between their performances for various
treatment lengths and viscoelastic layer thicknesses are studied.
QKNK léíáã~ä eóÄêáÇ a~ãéáåÖ mÉêÑçêã~åÅÉë
Table 5 shows some measures of the optimal open- and closed-loop performances of the
hybrid treatments, namely the first fivemodal damping factors (ζ i , i = 1, . . . , 5), the settlingtime ts and maximum amplitude ymax of the output y. It is observed that passive damping
factors of the ACL, AC/PCL and AC/PSOL treatments are rather similar, although that of the
first mode for the ACL is 15% superior than that for AC/PCL and AC/PSOL. The passive
damping of the APCL treatment is more effective for all modes (40% over AC/PCL and
AC/PSOL, 20% over ACL). This is why the open-loop ACL and APCL treatments lead to
smaller settling times. It can also be noticed that, although the damping of the second mode
is superior for the AC/PSOL (50% over ACL and AC/PCL, 13% over APCL), those of the
fourth and fifth modes are smaller for this treatment.
The maximum amplitude of the beam tip deflection is obviously identical for all the
treatments since it was initially fixed at hb/2 = 1.5mm for the evaluation of the perturbation
force. For the closed-loop hybrid damping, it is clear that the AC/PCL and AC/PSOL are
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 737
Table 5. Open- (passive) and closed-loop (hybrid) optimal performances of the hybrid
treatments (optimized to control the first three modes, Q = diag(1, 1, 1, 0, . . . , 0)).
ζ 1 (%) ζ 2 (%) ζ 3 (%) ζ 4 (%) ζ 5 (%) ts (sec) ymax (mm)
Base beam 0.10 0.10 0.10 0.10 0.10 30.0 1.50
ACL 2.91 2.35 2.13 2.77 2.74 0.80 1.50
Passive APCL 3.53 3.13 2.74 3.41 3.51 0.70 1.50
AC/PCL 2.53 2.30 1.63 2.24 2.51 0.90 1.50
AC/PSOL 2.53 3.54 2.18 1.79 2.00 0.90 1.50
ACL 6.79 3.54 2.22 2.77 2.74 0.35 1.41
Hybrid APCL 10.49 4.56 2.77 3.41 3.51 0.25 1.35
AC/PCL 20.83 5.84 1.74 2.24 2.51 0.10 1.15
AC/PSOL 19.38 4.77 2.67 1.79 2.00 0.12 1.22
much more effective in controlling the first mode actively. This is why the settling times for
these treatments are smaller than those for ACL and APCL. Similarly, one may observe that
the hybrid treatments performances allow the active reduction of the open-loop maximum
deflection amplitude by 6% for the ACL, 10% for the APCL, 23% for the AC/PCL and 19%
for the AC/PSOL.
It is worthwhile noting that Table 5 shows that AC/PCL leads to the best closed-loop
performance for the first two modes. However, one may use the sum ζ of passive ζp and
hybrid ζh damping factors, which are themselves the sum of the first three open- and closed-
loop modal damping factors, a measure of optimality, as in the preceding section. In this
case, optimal values of ζ are 19.90% for the ACL, 27.28% for the APCL, 34.95% for the
AC/PCL and 35.03% for the AC/PSOL. Hence, although AC/PCL is better in closed-loop
than AC/PSOL, the latter leads to better global performance. Moreover, one may notice that,
when accounting for both passive and hybrid damping factors, the performance of the APCL
treatment approaches those of AC/PCL and AC/PSOL.
It is also interesting to analyze the variation of the difference between the treatments
damping factors with the treatment length and viscoelastic layer thickness. These parametric
comparisons are performed in the following sub-section.
QKOK m~ê~ãÉíêáÅ `çãé~êáëçåë _ÉíïÉÉå eóÄêáÇ qêÉ~íãÉåíë
The global damping performances ζ = ζp + ζh of the ACL, APCL, AC/PCL and AC/PSOL
hybrid damping treatments are compared for the treatment length and viscoelastic layer
thickness ranges used in previous section. Although ACL and APCL treatments lead to
similar damping mechanisms, Figure 37 shows that the performance of the latter is superior
for all treatment lengths and viscoelastic layer thicknesses considered. This is mainly due to
the fact that APCL leads to a thicker constraining layer, thus better passive damping, and less
actuator controllability losses due to the viscoelastic layer. However, according to Figure 37,
the advantage of APCL over ACL does not exceed 8%.
Figure 38 presents the difference of the damping performances of AC/PCL and ACL
treatments according to the treatment length a and viscoelastic layer thickness hv . It showsthat the former is more effective than the latter for all values of a and hv . This figure alsoindicates that the advantage of the AC/PCL over the ACL increases with the treatment length
738 M. A. TRINDADE and A. BENJEDDOU
10−2
10−1
100
20
40
60
0
2
4
6
8
hv (mm) a (mm)
ζAPC
L−
ζAC
L (
%)
Figure 37. Influence of treatment length a and viscoelastic layer thickness hv on the difference of the
global damping performances ζ = ζp + ζh of APCL and ACL treatments.
and viscoelastic layer thickness. In fact, a maximum difference of 20.4% is obtained for thick
viscoelastic layer (hv = 2 mm) and long treatments (a = 70 mm). This is explained by the
fact that the ACL is effective only for very thin viscoelastic layers, whereas the AC/PCL
maintains a good performance over all the thickness range. The performances of AC/PSOL
and ACL treatments are compared in Figure 39. This figure indicates that the AC/PSOL
treatment, as the AC/PCL, is always more effective than the ACL one, in particular, for long
treatments and thick viscoelastic layers. A maximum difference of damping performance of
22.2% is obtained for a = 70 mm and hv = 0.2 mm, although it is almost constant in the
range 0.2 < hv < 2 mm. However, for long treatments, this difference decreases with the
reduction of the viscoelastic layer thickness. This is because, in the ranges 40 < a < 70mm
and 0.01 < hv < 0.03 mm, the performance of the ACL is almost optimal whereas that of
the AC/PSOL is rather minimal.
It is clear that hybrid treatments using the passive damping layers and piezoelectric
actuators separately (AC/PCL and AC/PSOL) are globally much more effective than the
others (ACL and APCL). It is thus worthwhile to compare the first two treatments. Figure 40
shows the difference between the damping sums ζ = ζp + ζh of AC/PSOL and AC/PCL
treatments. It can be observed that AC/PSOL is more effective than AC/PCL for short
treatments or relatively thick viscoelastic layers. This advantage is maximum (3.8%) for
a = 70 mm and hv = 0.2 mm. However, the AC/PCL is more effective than the AC/PSOL
for long treatments and very thin viscoelastic layers, since in this case the performance of
the first is almost optimum whereas that of the second is rather minimum. Indeed, AC/PCL
performance surpasses that of AC/PSOL by 9.5% for a = 70 mm and hv = 0.01 mm.
The results of the preceding and present sections confirm that hybrid treatments are
very effective to improve the modal damping of some selected modes, while keeping the
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 739
10−2
10−1
100
20
40
60
0
5
10
15
20
25
hv (mm) a (mm)
ζAC
/PC
L−
ζAC
L (
%)
Figure 38. Influence of treatment length a and viscoelastic layer thickness hv on the difference of the
global damping performances ζ = ζp + ζh of AC/PCL and ACL treatments.
10−2
10−1
100
20
40
60
0
5
10
15
20
25
hv (mm) a (mm)
ζAC
/PSO
L−
ζAC
L (
%)
Figure 39. Influence of treatment length a and viscoelastic layer thickness hv on the difference of the
global damping performances ζ = ζp + ζh of AC/PSOL and ACL treatments.
740 M. A. TRINDADE and A. BENJEDDOU
10−2
10−1
100
20
40
60
−10
−5
0
5
hv (mm) a (mm)
ζAC
/PSO
L−
ζAC
/PC
L (
%)
Figure 40. Influence of treatment length a and viscoelastic layer thickness hv on the difference of the
global damping performances ζ = ζp + ζh of AC/PSOL and AC/PCL treatments.
passive damping provided by the viscoelastic layer. The association of PCL treatments
with piezoelectric actuators can be effective, either when these actuators are bonded on the
viscoelastic treatment or directly on the structure. In the first case, the actuator must act on
the structure through the viscoelastic layer. It thus loses controllability when the thickness of
the viscoelastic layer grows, due to the reduction of the transmissibility between the actuator
and the structure. In the other case, the piezoelectric actuator can be bonded on the opposite
surface of the structure or between the viscoelastic layer and the structure so that the active
controller acts directly on the structure. Hence, the transmissibility is not much reduced by
the increase of the viscoelastic layer thickness. At the same time, the passive treatment keeps
a reasonable damping margin for the modes that are not actively controlled. On the other
hand, since the PCL treatment is optimal for rigid constraining layers, APCL, AC/PCL and
AC/PSOL lead to larger thickness and weight, compared to ACL treatment. However, since
AC/PCL and AC/PSOL are generally more effective, they allow the use of shorter passive
and active treatments with good performances.
5. CURRENT TRENDS AND FUTURE RESEARCH DIRECTIONS
Performances of hybrid vibration damping treatments presented here have been discussed in
the reviewed literature over the past decade, but using different analysis and control methods.
Hence, some primary conclusions about the attainable performances have been indicated,
but separately for one or more configurations. The present assessment study has the merit
of comparing the main hybrid configurations already studied with a unified approach that
uses the same FE analysis method and control strategy (optimal LQR algorithm). However,
this area is likely to continue attracting research interest since there is still need of more
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 741
reliable, robust and effective control systems. Moreover, as industrial applications are being
carried out, some new ideas to solve resulting challenging problems may come up. Current
researches in hybrid active-passive damping technologies are:
• Active Piezoelectric Damping Composites (Shields, Ro, and Baz, 1998) consisting of
piezoelectric fibers embedded in a viscoelastic matrix to control the composite damping
characteristics;
• Replacement of the elastic constraining layers by magnetic ones in segmented PCL treat-
ments such that attraction or repulsion between adjacent constraining layers improve pas-
sive energy dissipation (Ebrahim and Baz, 1998);
• Combination of Active-Passive Piezoelectric Networks and ACL or EACL treatments
(Tsai and Wang, 1997) to provide both piezoelectric, through shunting circuits, and vis-
coelastic passive damping;
• Application of electro-rheological fluids (Oyadiji, 1996) for hybrid active-passive treat-
ments;
• Combined use of shape memory alloys and constrained viscoelastic layers as an ACL
damping treatment for vibration control (Chen and Levy, 1999);
• Piezoelectric active control of already damped sandwich structures (Trindade, Benjed-
dou, and Ohayon, 2001a; 2001b) which consists in controlling the vibration of sandwich
structures with viscoelastic core through attached piezoelectric patches.
Moreover, research on the modeling and analysis of such multi-physics problems are still
needed. Future researches can be directed to:
• More refined models for piezoelectric and viscoelastic materials behavior, in particular
for temperature dependence and for laminated composite structures with embedded smart
materials;
• Application of piezoelectric-viscoelastic hybrid treatments, such as APCL, AC/PCL,
AC/PSOL, EACL, to more complex structures, in particular to two-dimensional struc-
tural components, such as plates and shells;
• Use of magneto/electrostrictive materials and shear piezoelectric actuators (Benjeddou,
Trindade, and Ohayon, 2000; Trindade, Benjeddou, and Ohayon, 1999) as active compo-
nents in hybrid active-passive treatments;
• Revision of model reduction techniques, already well-known in structural dynamics, to
multi-physics coupled problems as the hybrid active-passive ones;
• Control strategies adapted to hybrid active-passive coupled systems.
6. CONCLUDING REMARKS
A review of hybrid active-passive damping treatments, including geometric configurations,
modeling approaches and control algorithms, has been presented. It has been shown
that, through different combinations of viscoelastic layers, point or distributed sensors and
piezoelectric actuators, hybrid active-passive damping treatments provide reliable, robust,
adaptive and effective structural vibration control. Thereafter, using a unified finite element
model, able to represent sandwich damped beams with piezoelectric laminated faces, and
an iterative LQR optimal control algorithm, geometric optimization, through parametric
742 M. A. TRINDADE and A. BENJEDDOU
analysis, of the vibration damping of a cantilever aluminum beamwith ACL, APCL, AC/PCL
and AC/PSOL treatments was studied. Active control was designed to damp the first three
bending modes of the beam and geometric optimization was carried out to maximize the
sum of passive and hybrid modal damping factors of these modes. Present results have
confirmed that hybrid treatments are very effective in improving the modal damping of
the selected modes, while keeping the passive damping provided by the viscoelastic layer.
This parametric study was followed by a comparison of the performances of the different
treatments. It was shown that ACL and APCL treatments are effective only for very thin
viscoelastic layers, in order to prevent loss of controllability, although APCL treatment has
led to better performances than ACL one in both open- and closed-loop. Therefore, the use of
passive and active treatments separately (AC/PCL and AC/PSOL) has generally led to higher
performances. AC/PSOL was found to be more effective for relatively thick viscoelastic
layers whereas AC/PCL was more effective for very thin ones. Moreover, only APCL and
AC/PCL treatments allow different lengths of passive and active layers. It is therefore worthy
to note that other conclusions can be reached if different geometric and material properties
or control strategies are used for the hybrid configurations investigated in the present work.
Acknowledgments. This research was supported by the French Procurement Agency – Advanced Materials
Branch, under contract D.G.A./D.S.P./S.T.T.C./MA. 97-2530, which is gratefully acknowledged. The first
author acknowledges also the support of the Brazilian government (CAPES) through a doctoral scholarship,
grant no. BEX 2494/95-7.
NOTE
1. This reference could not be consulted but a consensus exists in the literature that it is the first work on
the ACL damping.
REFERENCES
Agnes, G.S. and Napolitano, K., 1993, ‘‘Active constrained layer viscoelastic damping,’’ in 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA, Reston, VA, USA, pp. 3499–
3506.
Azvine, B., Tomlinson, G.R., and Wynne, R., 1995, ‘‘Use of active constrained-layer damping for controlling resonant
vibration,’’ Smart Materials and Structures 4(1), 1–6.
Azvine, B., Tomlinson, G.R., Wynne, R., and Sensburg, O., 1993, ‘‘Vibration suppression of flexible structures using
active damping,’’ in Proceedings of the 4th International Conference on Adaptive Structures and Technologies,Breitbach, E.J., Wada, B.K., and Natori, M.C., eds., Technomic, Lancaster, PA, USA, pp. 340–356.
Badre-Alam, A., Wang, K.W., and Gandhi, F., 1999, ‘‘Optimization of enhanced active constrained layer (EACL) treat-
ment on helicopter flexbeams for aeromechanical stability augmentation,’’ SmartMaterials and Structures 8(2),182–196.
Bagley, R.L. and Torvik, P.J., 1983, ‘‘Fractional calculus – a different approach to analysis of viscoelastically damped
structures,’’ AIAA Journal 21(5), 741–748.
Baz, A., 1997a, ‘‘Boundary control of beams using active constrained layer damping,’’ Journal of Vibration andAcoustics119(2), 166–172.
Baz, A., 1997b, ‘‘Dynamic boundary control of beams using active constrained layer damping,’’ Mechanical Systemsand Signal Processing 11(6), 811–825.
Baz, A., 1997c, ‘‘Optimization of energy dissipation characteristics of active constrained layer damping,’’ Smart Mate-rials and Structures 6(3), 360–368.
Baz, A., 1998, ‘‘Robust control of active constrained layer damping,’’ Journal of Sound and Vibration 211(3), 467–480.
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 743
Baz, A. and Ro, J., 1993, ‘‘Partial treatment of flexible beams with active constrained layer damping,’’ in Recent De-velopments in Stability, Vibration and Control of Structural Systems, Guran, A., ed., Vol. AMD-167, ASME,
New York, pp. 61–80.
Baz, A. and Ro, J., 1994, ‘‘The concept and performance of active constrained layer damping treatments,’’ Sound andVibration Magazine pp. 18–21.
Baz, A. and Ro, J., 1995a, ‘‘Optimum design and control of active constrained layer damping,’’ Journal of Vibrationand Acoustics 117(B), 135–144.
Baz, A. and Ro, J., 1995b, ‘‘Performance characteristics of active constrained layer damping,’’ Shock and Vibration2(1), 33–42.
Benjeddou, A., 2000, ‘‘Advances in piezoelectric finite elements modeling of adaptive structural elements: A survey,’’
Computers & Structures 76(1-3), 347–363.
Benjeddou, A., 2001, ‘‘Advances in hybrid active-passive vibration and noise control via piezoelectric and viscoelastic
constrained layer treatments,’’ Journal of Vibration and Control 7(4), 565–602.
Benjeddou, A., Trindade, M.A., and Ohayon, R., 1999, ‘‘New shear actuated smart structure beam finite element,’’ AIAAJournal 37(3), 378–383.
Benjeddou, A., Trindade, M.A., and Ohayon, R., 2000, ‘‘Piezoelectric actuation mechanisms for intelligent sandwich
structures,’’ Smart Materials and Structures 9(3), 328–335.
Biot, M.A., 1955, ‘‘Variational principles in irreversible thermodynamics with application to viscoelasticity,’’ PhysicalReview 97(6), 1463–1469.
Brackbill, C.R., Lesieutre, G.A., Smith, E.C., andGovindswamy, K., 1996, ‘‘Thermomechanical modeling of elastomeric
materials,’’ Smart Materials and Structures 5(5), 529–539.
Chen, Q. and Levy, C., 1999, ‘‘Vibration analysis and control of flexible beam by using smart damping structures,’’
Composites Part B: Engineering 30(4), 395–406.
Chen, T. and Baz, A., 1996, ‘‘Performance characteristics of active constrained layer damping versus passive constrained
layer damping with active control,’’ in Smart Structures & Materials 1996: Mathematics and Control in SmartStructures, Varadan, V.V. and Chandra, J., eds., Vol. 2715, SPIE, Bellingham, WA, USA, pp. 256–268.
Crassidis, J., Baz, A., and Wereley, N., 2000, ‘‘H∞ control of active constrained layer damping,’’ Journal of Vibrationand Control 6(1), 113–136.
DiTaranto, R.A., 1965, ‘‘Theory of vibratory bending for elastic and viscoelastic layered finite-length beams,’’ Journalof Applied Mechanics 32, 881–886.
Dosch, J.J., Inman, D.J., and Garcia, E., 1992, ‘‘A self-sensing piezoelectric actuator for collocated control,’’ Journal ofIntelligent Material Systems and Structures 3(1), 167–185.
Douglas, B.E. andYang, J.C.S., 1978, ‘‘Transverse compressional damping in the vibratory response of elastic-viscoelastic-
elastic beams,’’ AIAA Journal 16(9), 925–930.
Dovstam, K., 1995, ‘‘Augmented Hooke’s law in frequency domain: A three dimensional, material damping formula-
tion,’’ International Journal of Solids and Structures 32(19), 2835–2852.
Ebrahim, A.K. and Baz, A., 1998, ‘‘Vibration control of plates using magnetic constrained layer damping,’’ in SmartStructures & Materials 1998: Passive Damping and Isolation, Davis, L.P., ed., Vol. 3327, SPIE, Bellingham,
WA, USA, pp. 138–158.
Enelund, M. and Lesieutre, G.A., 1999, ‘‘Time domain modeling of damping using anelastic displacement fields and
fractional calculus,’’ International Journal of Solids and Structures 36(29), 4447–4472.
Friswell, M.I. and Inman, D.J., 1998, ‘‘Hybrid damping treatments in thermal environments,’’ in Smart Materials andStructures, Tomlinson, G.R. and Bullough, W.A., eds., IOP Publishing, Bristol, UK, pp. 667–674.
Friswell, M.I. and Inman, D.J., 1999, ‘‘Reduced-ordermodels of structureswith viscoelastic components,’’ AIAA Journal37(10), 1318 – 1325.
Friswell, M.I., Inman, D.J., and Lam, M.J., 1997, ‘‘On the realisation of GHM models in viscoelasticity,’’ Journal ofIntelligent Material Systems and Structures 8(11), 986–993.
Golla, D.F. and Hughes, P.C., 1985, ‘‘Dynamics of viscoelastic structures – a time-domain, finite element formulation,’’
Journal of Applied Mechanics 52(4), 897–906.
Huang, S.C., Inman, D.J., and Austin, E.M., 1996, ‘‘Some design considerations for active and passive constrained layer
damping treatments,’’ Smart Materials and Structures 5(3), 301–313.
Inman, D.J. and Lam, M.J., 1997, ‘‘Active constrained layer damping treatments,’’ in 6th International Conference onRecent Advances in Structural Dynamics, Ferguson, N.S., Wolfe, H.F., and Mei, C., eds., Vol. 1, Southampton,
UK, pp. 1–20.
Johnson, A.R., Tessler, A., and Dambach, M., 1997, ‘‘Dynamics of thick viscoelastic beams,’’ Journal of EngineeringMaterials and Technology 119(3), 273–278.
744 M. A. TRINDADE and A. BENJEDDOU
Johnson, C.D., Keinholz, D.A., andRogers, L.C., 1981, ‘‘Finite element prediction of damping in beamswith constrained
viscoelastic layers,’’ Shock and Vibration Bulletin 50(1), 71–81.
Kapadia, R.K. and Kawiecki, G., 1997, ‘‘Experimental evaluation of segmented active constrained layer damping treat-
ments,’’ Journal of Intelligent Material Systems and Structures 8(2), 103–111.
Kerwin, Jr., E.M., 1959, ‘‘Damping of flexural waves by a constrained visco-elastic layer,’’ Journal of the AcousticalSociety of America 31(7), 952–962.
Lam, M.J., Inman, D.J., and Saunders, W.R., 1997, ‘‘Vibration control through passive constrained layer damping and
active control,’’ Journal of Intelligent Material Systems and Structures 8(8), 663–677.
Lam, M.J., Inman, D.J., and Saunders, W.R., 1998, ‘‘Variations of hybrid damping,’’ in Smart Structures & Materials1998: Passive Damping and Isolation, Davis, L.P., ed., Vol. 3327, SPIE, Bellingham, WA, USA, pp. 32–43.
Leibowitz, M.M. and Vinson, J.R., 1993, ‘‘On active (piezoelectric) constrained layer damping in composite sandwich
structures,’’ inProceedings of the 4th Int. Conf. Adaptive Struct. Tech., Breitbach, E.J., Wada, B.K., and Natori,
M.C., eds., Technomic Pub. Co., Lancaster, PA, USA, pp. 530–541.
Lesieutre, G.A., 1992, ‘‘Finite elements for dynamic modeling of uniaxial rods with frequency-dependent material prop-
erties,’’ International Journal of Solids and Structures 29(12), 1567–1579.
Lesieutre, G.A. and Bianchini, E., 1995, ‘‘Time domain modeling of linear viscoelasticity using anelastic displacement
fields,’’ Journal of Vibration and Acoustics 117(4), 424–430.
Lesieutre, G.A. and Govindswamy, K., 1996, ‘‘Finite element modeling of frequency-dependent and temperature-
dependent dynamic behavior of viscoelastic materials in simple shear,’’ International Journal of Solids andStructures 33(3), 419–432.
Lesieutre, G.A. and Lee, U., 1996, ‘‘A finite element for beams having segmented active constrained layers with
frequency-dependent viscoelastics,’’ Smart Materials and Structures 5(5), 615–627.
Lesieutre, G.A. and Mingori, D.L., 1990, ‘‘Finite element modeling of frequency-dependent material damping using
augmenting thermodynamic fields,’’ Journal of Guidance 13(6), 1040–1050.
Liao, W.H. and Wang, K.W., 1996, ‘‘A new active constrained layer configuration with enhanced boundary actions,’’
Smart Materials and Structures 5(5), 638–648.
Liao, W.H. and Wang, K.W., 1997a, ‘‘On the active-passive hybrid control of structures with active constrained layer
treatments,’’ Journal of Vibration and Acoustics 119(4), 563–572.
Liao, W.H. and Wang, K.W., 1997b, ‘‘On the analysis of viscoelastic materials for active constrained layer damping
treatments,’’ Journal of Sound and Vibration 207(3), 319–334.
Liao, W.H. andWang, K.W., 1998a, ‘‘Characteristics of enhanced active constrained layer damping treatments with edge
elements, part 1: finite element model development and validation,’’ Journal of Vibration and Acoustics 120(4),886–893.
Liao, W.H. andWang, K.W., 1998b, ‘‘Characteristics of enhanced active constrained layer damping treatments with edge
elements, part 2: systems analysis,’’ Journal of Vibration and Acoustics 120(4), 894–900.
Liu, Y. and Wang, K.W., 1998, ‘‘Enhanced active constrained layer damping treatment with symmetrically and non-
symmetrically distributed edge elements,’’ in Smart Structures & Materials 1998: Passive Damping and Iso-lation, Davis, L.P., ed., Vol. 3327, SPIE, Bellingham, WA, USA, pp. 61–72.
McTavish, D.J. and Hughes, P.C., 1993, ‘‘Modeling of linear viscoelastic space structures,’’ Journal of Vibration andAcoustics 115, 103–110.
Mead, D.J., 1999, Passive Vibration Control, John Wiley & Sons, New York.
Mead, D.J. andMarkus, S., 1969, ‘‘The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary
conditions,’’ Journal of Sound and Vibration 10(2), 163–175.
Oyadiji, S.O., 1996, ‘‘Applications of electro-rheological fluids for constrained layer damping treatment of structures,’’
Journal of Intelligent Material Systems and Structures 7(5), 541–549.
Park, C.H., Inman, D.J., and Lam, M.J., 1999, ‘‘Model reduction of viscoelastic finite element models,’’ Journal ofSound and Vibration 219(4), 619–637.
Plump, J.M. and Hubbard Jr., J.E., 1986, ‘‘Modeling of an active constrained layer damper,’’ in 12th InternationalCongress on Acoustics, Toronto, Canada, pp. #D4–1.
Plunkett, R. and Lee, C.T., 1970, ‘‘Length optimization for constrained viscoelastic layer damping,’’ Journal of theAcoustical Society of America 48(1), 150–161.
Preumont, A., 1997. Vibration Control of Active Structures: An Introduction, Kluwer, Dordrecht, The Netherlands.
Rongong, J.A., Wright, J.R., Wynne, R.J., and Tomlinson, G.R., 1997, ‘‘Modelling of a hybrid constrained layer/piezo-
ceramic approach to active damping,’’ Journal of Vibration and Acoustics 119(1), 120–130.
HYBRID ACTIVE-PASSIVE DAMPING TREATMENTS 745
Saravanos, D.A. and Heyliger, P.R., 1999, ‘‘Mechanics and computational models for laminated piezoelectric beams,
plates, and shells,’’ Applied Mechanics Review 52(10), 305–320.
Shen, I.Y., 1994, ‘‘Hybrid damping through intelligent constrained layer treatments,’’ Journal of Vibration and Acoustics116(3), 341–349.
Shen, I.Y., 1995, ‘‘Bending and torsional vibration control of composite beams through intelligent constrained-layer
damping treatments,’’ Smart Materials and Structures 4(4), 340–355.
Shen, I.Y., 1996, ‘‘Stability and controllability of Euler-Bernoulli beams with intelligent constrained layer treatments,’’
Journal of Vibration and Acoustics 118(1), 70–77.
Shen, I.Y., 1997, ‘‘A variational formulation, a work-energy relation and damping mechanisms of active constrained
layer treatments,’’ Journal of Vibration and Acoustics 119(2), 192–199.
Shields, W., Ro, J., and Baz, A., 1998, ‘‘Control of sound radiation from a plate into acoustic cavity using active
piezoelectric-damping composites,’’ Smart Materials and Structures 7(1), 1–11.
Sunar, M. and Rao, S.S., 1999, ‘‘Recent advances in sensing and control of flexible structures via piezoelectric materials
technology,’’ Applied Mechanics Review 52(1), 1–16.
Trindade, M.A., Benjeddou, A., and Ohayon, R., 1999, ‘‘Parametric analysis of the vibration control of sandwich beams
through shear-based piezoelectric actuation,’’ Journal of Intelligent Material Systems and Structures 10(5),
377–385.
Trindade, M.A., Benjeddou, A., and Ohayon, R., 2000a, ‘‘Finite element analysis of frequency- and temperature-
dependent hybrid active-passive vibration damping,’’ Revue Europeenne des Elements Finis 9(1–3), 89–111.
Trindade, M.A., Benjeddou, A., and Ohayon, R., 2000b, ‘‘Modeling of frequency-dependent viscoelastic materials for
active-passive vibration damping,’’ Journal of Vibration and Acoustics 122(2), 169–174.
Trindade, M.A., Benjeddou, A., and Ohayon, R., 2001a, ‘‘Finite element modeling of hybrid active-passive vibration
damping of multilayer piezoelectric sandwich beams. Part 1: Formulation and Part 2: System analysis,’’ In-ternational Journal for Numerical Methods in Engineering 51(7), 835–864.
Trindade, M.A., Benjeddou, A., and Ohayon, R., 2001b, ‘‘Piezoelectric active vibration control of sandwich damped
beams,’’ Journal of Sound and Vibration 246(4), 653–677.
Tsai, M.S. andWang, K.W., 1997, ‘‘Integrating active-passive hybrid piezoelectric networks with active constrained layer
treatments for structural damping,’’ in Active/Passive Vibration Control & Nonlinear Dynamics of Structures,Clark, W.W., Xie, W.C., Allaei, D., Namachchivaya, N.S., and O’Reilly, O.M., eds., Vol. DE-95/AMD-223,
ASME, New York, pp. 13–24.
van Nostrand, W.C. and Inman, D.J., 1995, ‘‘Finite element model for active constrained layer damping,’’ in ActiveMaterials & Smart Structures, Anderson, G.L. and Lagoudas, D.C., eds., Vol. 2427, SPIE, Bellingham, WA,
USA, pp. 124–139.
van Nostrand, W.C., Knowles, G., and Inman, D.J., 1993, ‘‘Active constrained layer damping for micro-satellites,’’ in
Dynamics & Control of Structures in Space II, Kirk, C.L. and Hughes, P.C., eds., WIT Press, Southampton,
UK, pp. 667–681.
Varadan, V.V., Lim, Y.-H., and Varadan, V.K., 1996, ‘‘Closed loop finite-element modeling of active/passive damping in
structural vibration control,’’ Smart Materials and Structures 5(5), 685–694.
Veley, D.E. and Rao, S.S., 1996, ‘‘A comparison of active, passive and hybrid damping in structural design,’’ SmartMaterials and Structures 5(5), 660–671.
Wang, G. andWereley, N.M., 1998, ‘‘Frequency response of beams with passively constrained damping layers and piezo-
actuators,’’ in Smart Structures &Materials 1998: Passive Damping and Isolation, Davis, L.P., ed., Vol. 3327,
SPIE, Bellingham, WA, USA, pp. 44–60.
Yellin, J.M. and Shen, I.Y., 1996, ‘‘A self-sensing active constrained layer damping treatment for a Euler-Bernoulli
beam,’’ Smart Materials and Structures 5(5), 628–637.
Yellin, J.M. and Shen, I.Y., 1998, ‘‘An analytical model for a passive stand-off layer damping treatment applied to an
Euler-Bernoulli beam,’’ in Smart Structures & Materials 1998: Passive Damping and Isolation, Davis, L.P.,ed., Vol. 3327, SPIE, Bellingham, WA, USA, pp. 349–356.
Yiu, Y.C., 1993, ‘‘Finite element analysis of structures with classical viscoelastic materials,’’ in 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA, La Jolla,CA, pp. 2110–2119.