hybrid active-passive damping treatments using ... · hybrid active-passive damping treatments...

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

http:\\www.sagepublications.com

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


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,


Figure 1. Hybrid active-passive damping configurations.


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.


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


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)


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).


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).


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


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


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)


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


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).


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-


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


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.


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


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).


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


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).


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


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


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.


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


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


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.


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


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


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 (%

)


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


10−2

10−1

100

20

40

60

5

10

15

20

25

30

hv (mm) a (mm)

ζ h (%

)


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 (%

)


for the three first modes (relative to open-loop) of the AC/PCL-treated beam.


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


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


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


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


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


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).


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 (%

)


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


10−2

10−1

100

20

40

60

5

10

15

20

25

30

hv (mm) a (mm)

ζ h (%

)


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 (%

)


for the three first modes (relative to open-loop) of the AC/PSOL-treated beam.


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


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.


102

103

−200

−180

−160

−140

−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



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


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


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


10−2

10−1

100

20

40

60

0

5

10

15

20

25

hv (mm) a (mm)

ζAC

/PC

L−

ζAC

L (

%)


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 (

%)


global damping performances ζ = ζp + ζh of AC/PSOL and ACL treatments.


10−2

10−1

100

20

40

60

−10

−5

0

5

hv (mm) a (mm)

ζAC

/PSO

L−

ζAC

/PC

L (

%)


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


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


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.


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.


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.


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.

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