Laura Martin-Gomez

Playing with frontiers: Tolkien-zines, between fandom and academia

Keywords:

Tolkien fandom has been active since the beginning of the 1960s and has produced a variety of fanzines, in multiple contexts. This paper explores the different types of Tolkien-dedicated fanzines in the USA and the UK in the first decades of Tolkien fandomit shows how "Tolkien-zines" have been influenced and inspired by science-fiction fandom. The analysis then focuses on a specific phenomenon observed in some fanzines, edited by fan communities: studying three examples, we observe how some amateur publications evolve and play with the frontier between fandom and academia.

Even though it seems to have taken a few years to take root in science-fiction fandom, Tolkien-related pieces began to appear and became common in fanzines after 1960, especially in the USA. Simultaneously, the first Tolkien-dedicated association, The Fellowship of the Ring, appeared in 1960 in California. It was founded by science-fiction fans, active in the Los Angeles Science-Fiction Society (LASFS), but showed a wish for a specific outlet for Tolkien fans. The group was ambitious and successful but didn't produce much or lived up to expectations 10 . Soon, some other initiatives popped up, as fanzines distributed through an APA 11 acknowledging the seniority of science-fiction fandomor as local groups, some of then becoming rapidly so big that they turned into national structures, like the Tolkien Society of America 12 .

In 1965 and the few years that followed, The Lord of the Rings suddenly reached fame in the USA, attracting the mass media's attention as an editorial battle was fought around the paperback editions 13 and as the counterculture movement and students adopted the book as their new favorite. Interestingly, the "Tolkien craze" didn't directly impact the UK, and the British fan groups were few and far apart and have not left written traces until 1968 14 . Still, by the mid-70s, there were a multitude of fan publications about Tolkien's works, coming from a range of people, groups and associations in both countries.

As imperfect as a categorization might be, it still provides an interesting insight that enlightens how differently Tolkien works have been received by his fans and readers 15 . Thus, the large corpus of fanzines that I used for my PhD research allows us to get a global view on types of amateur publications with Tolkien's works as the focus point. If we observe "Tolkienzines" as material objectsprinted pages stapled together and sent through mailand amateur productions, we can begin to distinguish several categories. Different criteria could be taken as a focus point in order to categorize these Tolkien dedicated fanzines 16 : who produces them, how they are spread or how they are technically made, being the most obvious. Given the importance of identity conveyed by fanzines as means of communication, we will focus on the different producers, that is to say who actually creates, prints and spreads "Tolkien-zines" 17 :

1. Personalzines: these zines are thus called because they come from a unique author-editor, who personally types, copies, staples and sends the fanzines to his readers. Sometimes, the fan gets some friend's help or has to rent the typing machine as he doesn't have one himself, but he has complete control of what gets printed in the pages and of the overall tone he wants to use. This doesn't mean that all the content is written by the editor himself, most of the time, the readers share their thoughts letters that are printed as "Letters of Comment" in a specific column or send articles that are printed as such by the editor if they correspond to what he expects for his fanzine.

In our corpus, these fanzines have existed in various contexts, but share the abovementioned common points. In the US in the 60s especially, there have been some personalzines at the very beginning of Tolkien fandom that were strongly linked to sciencefiction fandom and were mostly spread through the system of the APAs 18 . Interestingly, in the same period and place, some counterculture fanzines of the same types circulated that had a strong interest in Tolkien 19 . In the UK, some personalzines appeared on the fringe of the newly founded Tolkien Society at the beginning of the 70s: humouristic or polemical, their audience was mainly the same as the Tolkien Society's 20 . 2. Newsletters and bulletins: most of the time produced by a formal group like an association (see Mythprint, Green Dragon or Amon Hen 21 ), it sometimes also stemmed from the central member of a reading group (like News from Bree, edited by Pauline Johnson 22 ). These fanzines act as a means to connect people who don't see each other on a regular basis but do meet sometimes (like in reading groups for example) or as a way of fostering a feeling of belonging to the community through writingthe bulletin acts as a means to keep everyone up to date on publications and rumours 23 but also on the group's latest activities. 3. Letterzines: this type of fanzine could also fall into one of the two former categories, but in my view, is sufficiently idiosyncratic to deserve to be one on its own right. A letterzine is a fanzine that is almost entirely composed of readers' letters ("Letters of Comment") but compiled by an editor just like a personalzine. Our corpus included one such example, The Middle Earthworm: it appeared as a "correspondent" to a sister-fanzine in the USA (The Tolkien Journal) but even though the Tolkien Society of America soon "disappeared" (it merged with the Mythopoeic Society), it continued for some years to be printed regularly and independently by its editor Archie Mercer, and only contained letters from its readers. 4. Collaborative content fanzines: these fanzines are usually produced by formal groups or associations, sometimes there are also student fanzines linked to a university interest group. These publications are, by definition, the product of a collaboration between several people, some being usually in charge of writing some texts, others of typing, copying and sending them for example. Various columns usually mean various authors too. These fanzines, much more than the others, rely on the participation of its readers, not only for the letter column but also for its articles. This leads to an invariable litany coming up regularly in their pages: the editors are always in need for texts and call for help and more participation from their readers who seem very apathetic.

Thus, these different types of fanzines have clearly different dynamics, depending more or less on reader's participation or on the editor's will and time to create the material. Actually, one dynamic that seems to be frequent through all of Tolkien fandom, is the one where fanzines tend to evolve to such an extent that the term "fanzine" doesn't seem to apply to them anymore if it ever did. * Our study will now turn to three examples of publications that illustrate this trend among some Tolkien fan publications. We will focus on the Tolkien Journal, published by the Tolkien Society of America between 1965 and 1972 24 , Mythlore, published by the Mythopoeic Society since 1967 25 , and Mallorn, published by the Tolkien Society since 1970. All these publications came out as a printed product of non-professional people, committed to Tolkien's work, within the framework of a specific fan communityas such it would seem fair to call them fanzines.

Despite disappearing after a few years of existence, the Tolkien Journal showed some signs of evolution towards a scholarly journal quite soon. As for Mallorn, it actually became a peer-reviewed journal just this year and has also begun giving open access online to some of its earlier contents. It seems to have been following an example set by Mythlore, which was classified as journal in 1999. Thus, a few decades into existence, these publications are assimilated as academic or scholarly journals, rather than fan publications.

If the evolution towards a better editorship and quality makes sense given the long years of accumulated experience for these publications, what is actually interesting is to observe how, from the start, these publications didn't quite "fit in" the definition of a fanzines.

To begin with, the Tolkien Journal was edited by Richard Plotz, a New York high school student. From an educated and well-connected family, he advertised his newly founded Tolkien-group, initially dubbed "New York Tolkien Society", in the newspapers. From the start, as Dick Plotz didn't participate in science-fiction fandom and was actually unfamiliarized with it, he reached out to readers and fans in an uncommon process in fandom 26one of the consequences seems to have been the lightning speed at which the society grew. The Tolkien Journal was printed as a single page at first, but soon was several pages long and by the fifth issue included scholarly articles 27 . In the second issue, Dick Plotz, commenting on the project of a fanzines' list that a member of the society plans to compile, wrote: "The Tolkien Journal, although not strictly a fanzine, will be included" 28 . Thus, even though he launched and advertised his society and publication outside fandom, he also reached out to science-fiction fandom, as on second thought, while distinguishing his material from other fan publications. From its title to the range of advertisement, The Tolkien Journal definitely didn't "act as" a fanzine from the start. This was quickly confirmed by the tight links Dick Plotz established with the Ballantine Editions which published The Lord of the Rings in paperback, which led to some commercial ads being included in the publication. Also, Dick Plotz endorsed a role of "messenger" or "prescriptive" figure as he exchanged a few letters with Tolkien himself and only transcribed or talked about some extracts in the pages of the journal. On the other hand, the Tolkien Journal was clearly considered as the most-needed Tolkien-related material, as we can see with the huge number of members the Tolkien Society of America soon had. Designed as a non-fannish publication despite the amateurish means of printing, the Tolkien Journal gained a wide audience it probably wouldn't have had if it had merely circulated in the traditional science-fiction fandom networks, and at the same time it helped foster the first solid and wide Tolkien fan community that existed 29 .

On the other side of the country, Mythlore was published by the Mythopoeic Society, founded in California in 1967 by some students, and mainly led by Glen GoodKnight. The first issue appeared in 1969 when the society had already acquired some solid membership and some traditions of its own 30it was around 50 pages long. Like a fanzine, it included letters, columns, reports from meetings and some content articles and its production was clearly nonprofessional, but the goal seemed ambitious to say the least. Very soon it seems that the editors received mixed-feelings comments from the society's members who didn't quite know what to make of the publication, and Glen GoodKnight felt compelled to react on this matter: Thus, it seems that from the outset, the self-definition of this publication was an issuealbeit a voluntary one of sorts. Actually, this seems to reflect the mixed activities that the MythSoc has put in place since its inception: it's founded on both fannish practices (the first meetings combined costumes, games and quizzes on Middle-earth for example) and more scholarly ones (like reading meetings, with a specific defined programme). Yet, Glen GoodKnight opposes here seriousness and humour, as he does scholarship and fandoma partial view at best, but also a quite common one in Tolkien fandom 32 . While acknowledging the role of fans inside the society, Glen GoodKnight also seems to be paying more attention to the members who, while being capable of humour, are not used to fanzines and their specific discoursesdo not belong to fandom per se.

Turning to the UK, Mallorn is the main publication of the Tolkien Society, and its first issue appeared in 1970, just a few months after the society was created. It wasn't the first printed material associated with it though: Vera Chapman, nicknamed Belladonna Took, had already published a few issues of Belladonna's Broadsheet around the time the society was launched. In the first years of the society's existence, the issues of Mallorn were few and far apart, and the society's members didn't get a lot of news of what was going on in the society or as far as decisions from the committee were taken. As a society both connected to science fiction fandom in a waythrough the pages of a fanzine called Middle Earthworm edited by Archie Mercerand to less community-invested readers like Vera Chapman for example, the society had trouble finding its feet at firstmostly due to lack of experience among active members. As a result, personalzines discussing the society and its goal sprouted quickly, while simultaneously, a "bulletin", called Amon Hen, supposedly more reliable and frequent than Mallorn, was launched by the society.

With members from different backgrounds, from seasoned fans to newly "converted" readers who just discovered fandom, the Tolkien Society soon sought to find a balanced "middle-way" to include everyone. One of the moves this implied was the curbing of fan-speak, as in Amon Hen for example, where, following a poll inside the society, the editor writes: "Abbreviations in general, and 'zine' in particular, are disliked. Thus, we shall try to cut down on abbreviations in the future" 33 . This shows a trend towards a more "neutral" language, which somehow could be considered as paving the way for a more scholarly or at least less-connoted environment. As the publications became actually more regular, Amon Hen kept its role as a bulletin of information, while Mallorn increasingly took a more scholarly approach.

These three examples, despite their specificities, show a gradual professionalization of layout, general editing, tone and contents. Some contextual explanations come to mind quite easily: first, fanzines have gradually become more and more professional in their layout as computers and printers became more available to the general public, second, the accumulated As editor I feel it is both fish and fowl (you punsters lay off of this metaphor), and I like it that way personally. Why not (hopefully) have the best of both worlds. I like the serious scholarly articles to further discuss and probe the genre of literature we are interested in. But although we take our subject seriously, I trust we will not take ourselves as seriously. There is room for humor, whimsey, and a good laugh among friends. The letter column reflects a relaxedness I appreciate -keep it up. I imagine Mythlore would appear a rare bird indeed to those accustomed to serious professional journals, and an odd fish-to those engrossed in reading the publications of fandom 31 . experience of the editors and consequently the reputation of the publication helps improve the consistency and probably the quality demands. Moreover, these three publications were printed in specific context that we shouldn't neglect: all three were printed by associations, and they weren't their only means of communicationeach journal had a matching bulletin which in a sense could play the sociability role of fanzines 34 .

Still, the attraction towards a more serious and professional publication as far as layout goes does not necessarily imply a more academic approach to Tolkien, and yet that is what happened in each instance. Two different but complementary explanations come to mind: Tolkien readers, as well as science fiction readers of that period, were mostly educated and so it entails that part of the participants in fandom were to become academics, "Tolkien-zines" could be a sort of first glimpse or first trial at academic writing 35 . It also seems interesting to note the parallel between scholarly Tolkien fans who both enjoy fiction and dive into a serious and rigorous analysis of it with Tolkien's career himself, as a writer and as an academic.

*

The study of Tolkien-zines shows that there is an important diversity in the field, the fanzines ranging from politically oriented publication to very narrow interest-centered publications. In the US and to a lesser extent, in the UK, there were fanzines dedicated to Tolkien's work that took their roots and influence in very different environments, even though science fiction fandom was the predominant influence for most.

As the case study examples we commented upon illustrate, there seems to be a specific trend in Tolkien-themed publications that bridges gaps or attempts at defining new frontiers, between scholarship and fandom. As such, these publications actually to get the best of both worlds: passionate individuals, rigour and editing experience, scholarship without the burden of professional journalsespecially financial considerations, but perhaps also career interests for individual participants for example. Today, as Mythlore and Mallorn become open access peer-reviewed journals, they confirm their move towards an academic image. As it reflects a push for more access to knowledge in academia, it also looks like a faithful inheritance of their belonging to traditional fandom, in which participation and time was more valued than money. The Silmarillion (announced as soon as 1960, published posthumously in 1977). 24 As the TSA grew fast, Ed Meskys soon took over from Dick Plotz, but as he had health issues, the journals were joint editions for some issues before merging with Mythlore, just like the TSA merged with the MythSoc. 25 Let's just specify that this society is not solely dedicated to Tolkien but to "mythopoeic" authors, namely and mostly C. S. Lewis and Charles Williams for example. 26 Ed Meskys, a well-known science-fiction fandom and editor of Niekas, wrote at the time: "I understand that the founder is unfamiliar with fandom, and that several fans had seen an advertisement he placed in a newspaper or magazine and contacted him.", Niekas n°12, June 1965, p. 78. 27 Donald Reinken, "The Lord of the Rings: A Christian Refounding of the Political Order", reprinted in Tolkien Journal n° 5 (Vol. 2 n°3), Summer 1966, p. 4-9. Originally printed in Christian Perspectives, Winter 1966. 28 Tolkien Journal n°2, Autumn 1965, p. 3. 29 We mentioned above The Fellowship of the Ring which had been active a few years before in California, and was actually officially in existence when the TSA was launched. Despite being successful at first, it didn't really manage to create a viable independent fandom. 30 There were different branches which each had its own reading programme and meetings. Also "Bilbo's and Frodo's Birthday Party", first organized in 1967 by the MythSoc, soon became an important date in the calendar of Tolkien fans. 31 "Council of Elrond" (Glen GoodKnight's column), Mythlore Vol.1 n°4 (n°4), October 1969, p. 2. 32 The issue of tension between seriousness and humour and the distance to be put between oneself and one's passion to remain sociably acceptable has been studied by Christian Le Bart and other sociologists like Matthew Hills for example. 33 Amon Hen n°16, 1975, p. 2. This had already been remarked elsewhere, like in Gamma, a personalzine, where an American fan, Joan Brown commented on the word "zine" as "ugly" and that it shouldn't be used because "there must be more inoffensive words" (Joan E. Brown, Gamma n°3, n. d., p. 4). 34 As mentioned above, the Tolkien Society of America published the Tolkien Journal and soon added the bulletin called Green Dragon. The Mythopoeic Society began printing Mythprint more frequently than Mythlore as soon as 1970. As for the Tolkien Society, it tried several bulletin formats and titles before settling for Amon Hen in 1972. 35 Richard C. West began publishing Orcrist when he was a student at the University of Wisconsin: it began as a student publication mixing fannish and academic contents and Richard West later became one of the first and most respected bibliographer about Tolkien's work.

Other forms of amateur publications existed already, but Hugo Gernsback and his magazine Amazing Stories stand as the groundbreaking elements for science-fiction fandom.


Marcelo A Trindade

Ayech Benjeddou

Roger Ohayon

Piezoelectric Active Vibration Control Of Damped Sandwich Beams

HAL is

INTRODUCTION

Sandwich structures with embedded viscoelastic materials are widely used in aerospace and automotive industries due to their bene"cial performance in attenuating structural vibrations. The vibratory energy is dissipated through the shear strains induced in the soft viscoelastic layer by the relative displacements of the sti!er surface layers. It is well-known that the damping performance of such structures depends on the geometrical and material properties of each layer [START_REF] Mead | Passive Vibration Control[END_REF]. Also, the damping performance is generally limited by geometrical and weight constraints. Consequently, to improve the damping of some selected vibration modes, one may consider active means, such as piezoelectric actuation. Indeed, piezoelectric actuators associated with appropriate control systems are known to be sandwich beams, accounting for frequency dependence of viscoelastic materials properties, are the remaining originalities of this work.

SANDWICH BEAM SECOND ORDER MODEL CONSTRUCTION

An adaptive sandwich beam, composed of two laminated upper a and lower b surface layers, with, respectively, n and m elastic or piezoelectric sub-layers, and a viscoelastic core c is considered (Figure 1). The mechanical sti!nesses of the face sub-layers are composed of membrane and bending strains contributions (Euler}Bernoulli theory), while the core contains also a shear sti!ness (Timoshenko theory). Since this paper focuses on the control design and performance analysis, the formulation will be only brie#y presented here. Details can be found in reference [10].

FINITE-ELEMENT MODEL

The present FE model assumes Lagrange linear shape functions for the mean uN and relative uJ axial displacements of the surface layers and Hermite cubic ones for the transverse de#ection w. Electrical di!erence of potentials < IH (k"a, b; j"1,2,(nL , m L )) are considered only in the nL , m L piezoelectric sub-layers of the surface layer k and are assumed constant and uniform in the element (Figure 1). With these assumptions, the following elementary d.o.f. column vector q; e may be written:

q; e "col(uN , w , w , uJ , uN , w , w , uJ , < ?C ,2, < ?LK C , < @C ,2, < @K) C ),

where ( ) implies the x-space derivative, uN "(u ? #u @ )/2 and uJ "u ? !u @ . Discretization of the inertial, electromechanical and external virtual works of d'Alembert's variational formulation, leads to the equations of motion [10]

I H

M < e k j #M < e c q; G C # I H

K < e k j #K < e c q; e "F < e m , k"a, b; j"1,2,(n, m).

(

) 2
where ( K ) stands for the second time-derivative. M < e k j and M < e c are the elementary mass matrices of the k H th face sub-layer and core respectively. They are due to contributions related to translations in the x and z directions, and rotations. Lamination of the surface layers induce also translation}rotation coupling terms. The sti!ness matrices of the face sub-layers K < e k j may be decomposed into mechanical K < e k j m , piezoelectric K < e k j me and dielectric K < e k j e terms, such that K < e k j "K < e k j m !K < e k j me !K < e T k j me #K < e k j e , (3) whereas that of the core K < e c is composed only of a mechanical contribution. These matrices are de"ned in terms of the generalized displacements, uN , uJ and w, in reference [10]. F < e m is a mechanical point forces vector added a posteriori to the discretized system.

Since the mechanical and electrical d.o.f. are coupled statically only [10] and, decomposing the element d.o.f. vector q; e in mechanical d.o.f. q e and unknown (sensor) V e S and applied (actuator) V e A voltages so that q; e "col(q e , V e S , V e A ), system (2) may be condensed as

M e qK C #[(K e fm !K e fmeS K e \ 1 feS K e T fmeS )#K e c ] q e "F e m #F e e , (4)
where K e f represents the sum of all the sti!ness matrices of faces sub-layers. The applied voltages V e A provide an equivalent electrical load vector F e e "K e fmeA V e A , and the unknown potentials V e S are related to the mechanical d.o.f. q e by V e S "K e \ 1 feS K e T fmeS q e . Assembling the condensed system (4) for all elements produces

MqK #Dq #(K f #K c ) q"F m #F e , ( 5
)
where D is a viscous damping matrix added a posteriori and q is a velocity vector.

SECOND ORDER AUGMENTED ADF SYSTEM

It is known that the viscoelastic material properties depend on the excitation frequency and operating temperature. To account for the frequency dependence of the viscoelastic material, Lesieutre's ADF model [START_REF] Lesieutre | [END_REF] is used. However, the operating temperature is assumed constant and the self-heating of the viscoelastic material is neglected. Temperature variation e!ects on viscoelastically damped beams have been investigated elsewhere [START_REF] Trindade | [END_REF]. By supposing a frequency-independent Poisson ratio, discretized equations of motion (5) can be rewritten in the frequency domain as

+! M#j D#[K f #GH( ) K c ], q "F m #F e , (6)
where GH( ) is the complex frequency dependent shear modulus of the viscoelastic layer and ( ) stands for Fourier-transformed variables. Including n series of ADF dissipative d.o.f. q d i (i"1,2, n) and re-transforming to the time domain leads to the following augmented system [3]

M q G #D q #K q "F m #F e , (7)
with

M " M0 00 , D " D0 0D dd , F m " F m 0 , F e " F e 0 , K " K f #K c K qd K T qd K dd , q "col(q, q d 1 ,2, q d n ),
where

D dd "G diag ((C / ) 2(C L / L ) ), K dd "G diag (C 2C L ), K qd "[!K c T2!K c T] and K c "G K c , G "G (1# G G ), C G "(1# G G )/ G . Material parameters G and ( G , G )( i"1,2
, n) are evaluated by curve-"tting of the measurements of GH( ) [3]. is a diagonal matrix of non-vanishing eigenvalues of K c and T is the corresponding eigenvectors matrix.

STATE SPACE MODEL CONSTRUCTION

To apply the augmented "nite-element model presented above to an optimal control design, system (7) needs to be transformed into a state space form. Therefore, a state vector x is formed by the augmented vector q and the time derivative of the mechanical d.o.f. vector q . The time derivatives of the dissipative d.o.f. q d i are not considered since these variables are massless. From equation (7), this state is subjected to mechanical and electrical loads F m and F e , where the latter are proportional to the voltages applied to the piezoelectric actuators.

STATE SPACE AUGMENTED ADF SYSTEM

The second order system (7) may be transformed to the following state space one:

x "Ax#Bu#p, y"Cx,

where each column of the control input matrix B represents the piezoeletric actuation loads distribution for a unitary voltage and the control input u is a column vector formed by the voltages V applied to the actuators. The perturbation vector p is the state distribution of the mechanical loads F K and the output vector y is, generally, composed of the measured quantities, written in terms of the state vector x through the output matrix C. The system dynamics are determined by the square matrix A. Matrices and vectors of system (8) are

A" 00 2 0I ( /C )T T ! I0 0 $ \ 0 ( L /C L )T T 0 ! L I0 !M\ 1 (K f #K c ) M\ 1 K c T 2 M\ 1 K c T !M\ 1 D , x" q q , p" 0 M\ 1 F m , B" 0 M\ 1 FH e , C"[C qN C qR ], u"V A , (9)
where FH e is the voltage factored-out electrical loads vector and C qN and C qR are the output matrices relative to the augmented d.o.f. vector q and mechanical d.o.f. derivatives q , respectively. It is worthwhile noting that, although the unknown electrical d.o.f. (sensor voltages) were condensed in the FE model, they may be considered as measured output by using the relation between V S and q to de"ne the output matrix C.

COMPLEX MODAL REDUCTION

The dimension of the state space system (8) is too high for use in an optimal control design. Thus, a complex-based modal reduction is applied to this system. It neglects the contributions of the viscoelastic relaxation modes and the elastic modes related to eigenfrequencies out of the frequency range considered. Hence, the eigenvalues matrix and left T l and right T r , eigenvectors of equation ( 8) are "rst evaluated from AT r " T r , A T T l " T l , (10) so that T T l T r "I, then decomposed as

" P 00 0 L 0 00 B , T l "[T lr T ln T ld ], T r "[T rr T rn T rd ], ( 11
)
where P is the retained elastic eigenvalues matrix and T lr and T rr are its left and right corresponding eigenvectors matrices respectively. L and B correspond to the neglected elastic and relaxation eigenvalues respectively. T ln , T rn , T ld and T rd are their corresponding left and right eigenvectors. Consequently, the state vector is approximated as x+T rr x r and, using equations ( 10) and (11), system (8) may be reduced to

x r " P x r #T T lr Bu#T T lr p, y"CT PP x P . ( 12
)

REAL REPRESENTATION OF THE COMPLEX MODAL REDUCED MODEL

The main disadvantage of the reduced state space system [START_REF] Friot | 2CKC Colloque GDR Vibroacoustique[END_REF] is that its matrices are complex. Therefore, one may prefer to transform it to a real system before using it in the control design. Fortunately, since all overdamped (relaxation) modes were neglected, all elements of system (12) are composed of complex conjugates, such that

r " \ H M H \ , T T lr B" $ H H $ , T T lr p" $ H H $ , CT rr "[2 H M H 2], (13)
where H ( j"1, 2 , r) are the retained elastic eigenvalues and M H their complex conjugates. To construct a real representation of the state space system [START_REF] Friot | 2CKC Colloque GDR Vibroacoustique[END_REF], one may use a state transformation x; "T c x r , where T c is de"ned as [START_REF] Friot | 2CKC Colloque GDR Vibroacoustique[END_REF] T c "

\ !j/2Im( H ) j/2Im( H ) \ \ !j (Re( H )/Im( H )) #j (Re( H )/Im( H )) \ , (14)
so that the real state space system equivalent to equation ( 12) is

x; "A < x; #B < u#p; , y"C < x; . ( 15
)
where

A < "T c P T\ 1 c " 0I \\ !" H " 2Re( H ) \\ , B < "T c T T lr B" $ Im( H )/Im( H ) $ Re( H )#Im( H )R e ( H )/Im( H ) $ , p; "T c T T lr p" $ Im( H )/Im( H ) $ Re( H )#Im( H )R e ( H )/Im( H ) $ , C < "CT rr T\ 1 c "[2!2[Re( H )Re( H )#Im( H )Im( H )]22Re( H ) 2].
It is clear that the eigenvalues of the real matrix A < are exactly the elements of P . In form (15), the new state variables x; represent the modal displacements and velocities.

CONTROL DESIGN

The analysis of the LQR algorithm for the hybrid piezoelectric-active viscoelastic-passive vibration control was presented in previous papers [3,10]. Here, the objective is to analyze and compare the hybrid damping performance of other control algorithms, which is done for the "rst time. Hence, in this section, three control strategies are brie#y presented, namely LQR, LQG and derivative feedback. It is clear that, for practical applications, these algorithms must be used in an iterative form to account for maximum control voltage, since piezoelectric actuators can be depoled by high oscillating voltages.

CONSTRAINED OPTIMAL CONTROL STRATEGY

The LQR optimal control algorithm consists of minimizing the functional J"

(x; T Qx; #u T Ru)dt. ( 16
)
subjected to linear constraints (15). This minimization yields a linear full-state feedback control law u"!K g x; , where the control gain matrix K g "R\ 1 B < T P is evaluated by solving for P the following algebraic Riccati equation

A < T P#PA < !PB < R\ 1 B < T P#Q"0. (17)
Evidently, the performance of this controller depends on the state Q and input R weight matrices. While Q de"nes the relative weight of each state variable, R de"nes the relative weight of each actuator voltage. The latter may then be adjusted to limit the maximum voltage. The iterative algorithm developed and presented in reference [START_REF] Trindade | [END_REF] is used here to perform such "tting automatically. However, the voltage limitation leads to a perturbation-dependent performance of the controller [START_REF] Trindade | [END_REF].

OPTIMAL FULL-STATE OBSERVER AND CONTROLLER

The full-state feedback control using the LQR algorithm is clearly only feasible when all state variables can be measured. Since, generally, this is not the case, one must estimate the state variables using the measured ones. For that, let one de"ne the Luenberger state observer [START_REF] Preumont | Vibration Control of Active Structures: An Introduction[END_REF] as

x "A < x #B < u#p; #K e (y!C < x ), x (0)"0, ( 18
)
where x is the estimated state vector and K e is the observer gain matrix. The Luenberger observer simulates real system (15) and penalizes the di!erence between the measured output y and the estimated output C < x . Input w and output v noise contributions are added, respectively, to state excitation p and output measurement y in equation ( 15). Hence, replacing y"C < x; #v in equation ( 18), then subtracting the resulting equation from the state space system (15a), with added noise term w, leads to

e "(A < !K e C < )e#w!K e v, (19)
where e"x; !x states for the observation error and has initial value e(0)"x; (0). One notices, from equation ( 19), that the observation error e is asymptotically stable if the eigenvalues of A < !K e C < , also named observer poles, have negative real parts. It can be also shown [START_REF] Preumont | Vibration Control of Active Structures: An Introduction[END_REF] that if the system is completely observable, one might place the observer poles arbitrarily far in the complex left mid-plan to obtain fast error convergence, thus fast state estimation. However, this yields to large K e elements that may amplify the measurement noise v. Hence, a compromise between noise ampli"cation and state estimation velocity must be found. The well-known Kalman}Bucy Filter (KBF) allows the observer to evaluate gain matrix K e for white input w and output v noises [START_REF] Preumont | Vibration Control of Active Structures: An Introduction[END_REF]; that is, for expectations E[w]"0 and E[v]"0. Then, the optimal gain K e is obtained through minimization of the following quadratic cost function

J C "h T P e h (20)
for any arbitrary vector h and P e "E[ee T ]. The optimal observer covariance matrix P e is the solution of the following algebraic Riccati equation

A < P e #P e A < T !P e C < T V\ 1 C < P e #W"0. (21)
Here W and V are the input w and output v noises covariance matrices. The optimal gain K e is then de"ned as

K e "P e C < T V\ 1 . ( 22
)
From equation ( 19), one notices that the matrix V penalizes the observer gain to limit the ampli"cation of the output noise through the term K e v. Whereas, for noisy inputs, the matrix W increases the observer gain to damp quickly the error this noise produces.

Assembling system (15a), with added noise terms and for u"!K g x , and equation ( 19) yields the coupled system

x; e " A < !B < K g B < K g 0A < !K e C < x; e # p; 0 # I0 I !K e w v , (23)
where equation ( 15b) is still valid. This combination of an optimal LQR controller with a KBF optimal observer is known as the LQG algorithm. Owing to the block triangular form of this system, one may design the LQR optimal controller alone, then use it to design the optimal observer. However, the stability of coupled system (23) is evidently dependent on the stability of both sub-systems.

DERIVATIVE FEEDBACK CONTROLLER

An alternative to the complexities involved in the calculations of an optimal observer and controller is a simple derivative feedback. Considering that the output vector y can be measured, one may feed it back after ampli"cation to the actuator as a control voltage u"!K d y. Notice that the output vector is a linear combination of the state variables (y"C < x; ), which are themselves a combination of the "nite-element model variables q and their time derivatives. Hence, using the decomposition of the output matrix C into C qN and C qR (cf. equation ( 9)), leads to

y"C < x; "C < T c T T lr x"C qN q #C qR q . ( 24
)
To get a derivative control, the output is set to a combination of time derivatives only, so that C qN "0. Transforming back the state space system (8) to a second order one, using equation ( 9) together with u"!K d C qR q , one notices that such a derivative control is equivalent to the substitution, in system (7), of the following electrical loads vector

F e "!FH e K B C qR q . ( 25
)
Consequently, when moved to the l.h.s. of equation (7), this leads to a modi"cation of its damping matrix which becomes

D H" D#K B FH e C qR 0 0D dd . ( 26
)
It is clear that the positive de"niteness of the additional damping matrix K B FH e C qR , provided by the active controller, is a consequence of the relation between input FH e and output C qR vectors. Since FH e and C qR are relative to the mechanical d.o.f. that are a!ected by the piezoelectric actuator and those that contribute to the output, respectively, the increase of modal damping is dependent on the relative positions of the piezoelectric sensor and actuator.

NUMERICAL RESULTS

In this section, the piezoelectric active control of a cantilever sandwich beam, passively damped by its viscoelastic core, is studied. Figure 2 shows the con"guration of the cantilever

(mm) 3 h N "0)5 h T "0)1 Width, b (mm) 25 25 25
sandwich beam. Its material and geometrical properties are given in Table 1. The objective is to use the piezoelectric patches to increase actively the damping of the "rst bending modes of the passively damped beam. First, a parametric analysis of the active and passive control performances, using LQR control algorithm, is performed to "nd the optimal geometric con"guration. Then, derivative feedback and LQG controllers are applied to this optimal con"guration to compare the three controllers' performances.

The frequency dependence, in the range 20}5000 Hz, of the ISD112 viscoelastic material at 273C is represented by a three-series ADF model, with the parameters. G "0)50 MPa, "[0)746, 3)265, 43)284] and "[468)7, 4742)4, 71532)5] rad/s. These optimal ADF parameters yield a good approximation to the measured material master curves in the frequency range of interest as shown in Figure 3. Indeed, both elastic modulus G and loss factor evaluated using ADF parameters (dashed lines) match well with measured ones (solid lines). The model reduction presented previously is performed by keeping the "rst "ve bending eigenmodes of the damped sandwich beam. Since the corresponding complex reduced model has been validated in a previous paper [10], the focus here is on its equivalent real representation which is used for the "rst time. However, from section 3.3, one notices that their dynamical behavior must be equivalent. Figure 4 shows the frequency response function between the perturbation force and piezoelectric sensor using the full model (8) and the real reduced-order model (15). One notices that responses match well in the frequency range of interest.

LQR OPTIMAL CONTROL OF THE SANDWICH DAMPED BEAM

To improve actively the damping of the "rst three bending modes, the piezoelectric actuator bonded on the upper surface of the sandwich beam is combined to an iterative LQR algorithm [START_REF] Trindade | [END_REF], with parameters Q"diag(1, 1, 1, 0,2, 0) and R" I. The factor is evaluated automatically in the LQR algorithm to limit the control voltage to 250 V, leading to a maximum electrical "eld of 500 V/mm in the piezoelectric actuator. The piezoelectric sensor, although not used in this case, is preserved for future comparisons. A perturbation transverse force is applied to the beam tip, whose de#ection is measured and restrained to 1)5mm.

A parametric analysis of the passive damping, provided by the viscoelastic layer, and hybrid damping, which is the passive one increased by the active controller, is performed. This is achieved by varying the piezoelectric patch length in the range 20}70 mm and the viscoelastic core thickness in the range 0)01}2 mm. To guarantee adequate levels of damping for the "rst three bending modes, parameters optimization is based on the sum of these three modal damping factors. Figure 5 shows that increase of piezoelectric patch length has little e!ect in passive damping. In fact, due to an augmentation of the surface layer sti!ness, small improvement in the second and third modal damping is obtained. However, this does not enhance the passive damping of the "rst mode and is not observable in Figure 5. This "gure shows also that the passive damping is optimal for relatively thin viscoelastic layers (h T "0)1 mm) and decreases for both very thin (h T "0)01 mm) and thick (h T "2 mm) layers.

The sum of the "rst three closed-loop modal damping factors of the sandwich beam is presented in Figure 6 for several piezoelectric patch lengths and viscoelastic core thicknesses. It appears that the hybrid damping performance is optimal for long actuators (a"70 mm) and relatively thin viscoelastic cores (h T "0)1 mm), although it is adequately e!ective in the range 40(a(70 mm and 0)1(h T (2 mm. Nevertheless, the damping gain provided by the active controller, de"ned as ?

" F / N !1, is optimal for very thin viscoelastic cores, as shown in Figure 7. This is mainly due to the fact that passive damping is not e!ective in that case. However, this quantity gives a good measure of the advantage of considering an active controller in addition to a passive damping.

To yield both hybrid and passive performant damping properties, optimal parameters are set to maximize the sum of open-and closed-loop damping factors, which are already the sum of the corresponding "rst three modal dampings. The optimal damping performances of the open-and closed-loop sandwich beam are presented in Table 2. The latter shows that, for optimal parameters, that is for a"70 mm and h T "0)1 mm, the "rst mode damping is increased by 400% and the non-controlled fourth and "fth modes are not excited by the controller. The damping factors of the second and third modes are just slightly improved by the active control.

For the optimal parameters, frequency-and time-domain responses of the sandwich beam are evaluated in both open-and closed-loops. Hence, the frequency response function between the transverse excitation force and the tip de#ection of the beam is evaluated and shown in Figure 8. It shows that the active controller produces a decrease of 14 dB in the amplitude of the "rst resonance, compared to the passive case. Moreover, it preserves or enhances the passive damping of the other modes. It should be noticed, in Figure 8, that the last four modes considered are highly damped, as also shown in Table 2, so that they are less observable. This passive damping performance prevent instabilities due to spillover.

The open-and closed-loop transient responses of the sandwich beam tip de#ection are presented in Figure 9. It should be noticed that the closed-loop response settles much faster than the open-loop one. In fact, as shown in Table 2, the settling time t Q and maximum amplitude y max of the closed-loop beam transient response are reduced by 83% and 29%, respectively, compared to those in the open loop. The control voltage applied to the piezoelectric actuator to obtain this performance is shown in Figure 10. It is clear that the control voltage subsides as fast as the closed-loop response. Also, as required by the iterative optimal control algorithm, the control voltage is limited to 250 V.

DERIVATIVE CONTROL OF THE OPTIMAL BEAM CONFIGURATION

The main drawback of the LQR algorithm is the need to measure all state variables to construct the full-state feedback. Thus, in this section, the much simpler approach of derivative feedback is applied to the optimal case of the cantilever sandwich beam with bonded piezoelectric actuator and sensor of Figure 2, that is for a"70 mm and h T "0)1 mm. To this end, two measured outputs are considered, namely the time-derivative of the beam tip de#ection w R * and that of the voltage < Q 1 of the piezoelectric sensor bonded on the bottom surface of the sandwich beam (Figure 2).

¹ip de-ection velocity feedback

First, a control voltage proportional to the time derivative of the sandwich beam tip de#ection w R * is considered, such that < "!K B w R * . It is clear from equation ( 26) that higher control gains lead to higher additional damping factors and, provided that the actuator/sensor combination increases the damping of a given set of modes, the total damping will increase with the control gain. However, since the control voltage is proportional to the control gain, it is also clear that large control gains yield to high control voltages. Hence, these should be limited to the maximum voltage that can be applied to the piezoelectric actuator.

Figure 11 shows the root locus between a point velocity sensor on the tip of the beam and the piezoelectric actuator. In this "gure, the increase of the control gain K

B moves the open-loop poles, ;, toward the open-loop zeros, ᭺ . Moreover, the modal damping Im(s)/Re(s) increases when the poles move away from the imaginary axis Re(s)"0. ) } ) }, 302; )))))), 605.

Consequently, from Figure 11, one notices that this control law allows one to increase greatly the "rst and fourth modal dampings and, for large control gains, that of the second mode. However, it reduces the third and "fth modal dampings, for all control gains, such that their destabilization is only prevented by their passive damping. Thus, one notices that the viscoelastic passive damping increases the stability margins of the modes excited by the active control. The closed-loop poles, #, for the maximum control gain considered (K B "605) are also presented in Figure 11. The control gain K B is limited to prevent voltage saturation in the piezoelectric actuator (in this case, 250 V). One notices that, although from Figure 11 this control law may improve the second modal damping, the voltage limitation leads to its decrease.

The responses of the "rst "ve bending modes, in the frequency domain, are presented in Figure 12, for three control gains 137, 302 and 605. One notices that, as observed previously, the "rst and fourth modes are well damped, compared to the open-loop response (K B "0), whereas the other modes are excited for the three control gains, as expected from the analysis of Figure 11. The additional modal dampings provided by the active controller are presented in Table 3 (Gain w R *

). This table shows that the modes 2, 3 and 5 are excited by the active controller. That is, their modal dampings are reduced compared to the passive damping. However, as observed also in Figure 11, the passive damping due to the The open-and closed-loop time-domain impulsive responses of the beam tip de#ection, loaded by the perturbation transverse force, are presented in Figure 13, for the three control gains considered. One notices that the settling time of the transient response of the beam tip decreases with the control gain. For K B "605, the derivative feedback performances are similar to those obtained with the LQR controller (Figure 9). However, to obtain these performances while respecting the piezoelectric actuator voltage limitation (250 V), the beam tip de#ection is limited here to 0)36 mm, that is, 24% of that considered for the LQR controller (1)5 mm). Figure 14 shows the actuator control voltage for each control gain. It can be seen that, for K B "605, the control voltage is, indeed, limited to 250 V. In addition, it presents fast oscillations representing the voltage for actively controlling the fourth mode.

Sensor voltage rate feedback

Here, the piezoelectric sensor, bonded on the bottom surface of the sandwich beam (Figure 2), is used in the control loop. In this case, the time derivative of the voltage < 1 , induced in the sensor by the beam de#ection, is measured, ampli"ed and fed back to the actuator. Consequently, the control voltage is given by < "!K B < Q 1 . In the previous case, the voltage needed to cancel the "rst bending mode was opposite to the beam tip de#ection, so that a negative-control feedback led to vibration damping. However, in the present case, the beam de#ection induces opposite voltages in the piezoelectric patches. Thus, the control gains must be negative. Indeed, the application of a control voltage of the same sign as the sensor voltage leads to a bending cancellation. Figure 15 shows the root locus between the piezoelectric sensor and actuator bonded on the opposite surfaces of the sandwich beam, for control gains varying in the range [0,!R ]. One may notice that all modes are damped for small control gains K B , whereas for large ones, the "rst and "fth modes are destabilized. The poles and zeros corresponding to the second mode almost cancel each other, so that damping increase may only be small. The closed-loop poles, #, for the highest control gain K B "!13)5;10\ are also presented in Figure 15. Notice that those corresponding to the "fth mode are in the real axis Im(s)"0, since this mode is overdamped by the controller.

In Figure 16, the frequency response of the sensor voltage, limited to the "rst "ve bending modes, is presented for three control gains !4)5;10\, !9)0;10\ and !13)5;10\. As expected from the analysis of Figure 15, all modal damping is increased. Figure 16 shows that the hybrid controller eliminates the resonance of mode 5 and highly attenuates those of the modes 3 and 4. One may notice that, whereas the controller proportional to the tip velocity improves the "rst mode in spite of the others, the present one increases all modal damping (Table 3). This may be explained by the evident major contribution of the "rst mode to the beam tip de#ection. This dependence of the damping performance on the relative positions of actuator and sensor is the main disadvantage of the derivative controllers. In fact, in both cases studied here, the control performance of the "rst mode is rather limited when compared to that obtained with the LQR algorithm.

The open-and closed-loop time-domain impulsive responses of the beam tip de#ection are presented in Figure 17. One notices that the transverse vibrations of the sandwich beam are damped only slightly faster then for the velocity feedback controller (cf., Figure 13). The similarity of the settling performances of Figures 13 and17 is mainly due to the similarity of the closed-loop damping factors of the "rst mode for both controllers (Table 3), since the tip response is almost only composed of this mode contributions. Figure 18 shows the control voltages imposed to the piezoelectric actuator for the three control gains. As for the previous case, the voltage is below the limit of 250 V. One can also see, from Figure 18, that the fast oscillations of the control voltage are damped more quickly than for the previous controller. This may be due to the fact that the higher-frequency modes are better damped in this case. This section has shown that derivative control, although simple to implement, is much less e!ective than LQR. In addition, for non-collocated sensor and actuator (section 5.2.1), this algorithm leads to a decrease in the damping performance of some modes. Nevertheless, in practical active control design, it is usually di$cult to avoid excitation of some modes, due to modelling errors or noise and delay in control electronics. Hence, the destabilization e!ect of tip de#ection velocity feedback shows how the presence of some passive damping is important to prevent total destabilization of the excited modes. On the other hand, although the almost collocated derivative control of section 5.2.2 does not excite the modes considered, it provides also very poor damping performance. This is due to the fact that optimization of the damping of some selected modes is very dependent on the relative actuator/sensor position and the passive damping.

LQG CONTROL OF THE OPTIMAL BEAM CONFIGURATION

In this section, the optimal full-state observer presented previously (section 4.2) is applied to the control of the sandwich beam of Figure 2 with the optimal parameters a"70 mm and h T "0)1 mm. The interest of considering an LQG controller is to obtain the control parameterization, robustness and performance of LQR while measuring only the output y, instead of all state variables. Since one may design the LQR controller alone and then couple it with a Kalman Filter, the approach of section 5.1 will be used to evaluate the control gain K g and then, for given input W and output V noise covariances (scalars in this case), system (23) is constructed. The main di$culty introduced by the observer is the coupling between its own dynamics with that of the mechanical system. Moreover, since the initial conditions of all state variables are not known, the initial observer error is e(0)"x; (0). Consequently, as long as the observer error does not disappear the controller will not be able to control the beam. In the opposite, if the initial state x; (0) is known, the initial observer error vanishes (e(0)"0) and the LQG closed-loop response is equivalent to that of the LQR.

Considering the practical case, where the initial state is unknown, the responses of the system, uncontrolled and controlled by LQR and LQG controllers, are compared for several output noise covariances V. The input noise covariance is "xed to W"10\ 5 . Figures 19 and20 present the transient output response and control voltage for V"1.

Figure 19 shows that the LQG open-and closed-loop responses are the same, meaning that the LQG controller is not e!ective for this output noise. This may be explained by the fact that, since V is relatively large, corresponding to a noisy output measure, the observer gain remains small to prevent injection of this noise in the system. Consequently, the observer error is damped more slowly than the LQR closed-loop system. Thus, the LQG control voltage u"!K g x "!K g (x; !e) is almost nil, as shown in Figure 20, and does not a!ect the system. Reducing the output noise covariance to V"10\ 8 , the observer error converges quickly to zero and the state is estimated before being damped, as shown in Figure 21. However, for a performance similar to that of the LQR, the control voltage of the LQG controller is smaller during the initial instants and then larger than that imposed by the LQR one (Figure 22). This is due to the time needed for the observer to well estimate the state and inject it in the control. Hence, as long as the error does not vanish, the estimated state x is small compared to the real one x; , such that the control voltage u"!K g x is also small. Then, x converges to x; and the LQG control voltage increases to damp the system response. This delay e!ect is quite advantageous in this particular case since the maximum control voltage is reduced (Figure 22) for a similar damping performance.

To accelerate the state estimation, the output noise covariance is reduced to V"10\ 10 . However, as shown in Figures 23 and24, although the state estimation is faster, the control performance decreases and the maximum control voltage exceeds the actuator saturation voltage of 250 V. In fact, the maximum output amplitude is smaller in this case (y max " 1)40 mm against y max "1)43 mm for V"10\ 8 ), although still larger than that of the LQR controller (y max "1)10 mm). Nevertheless, the settling time (t Q "0)14 s) is higher than that for V"10\ 8 , which is equal to that of LQR (t Q "0)12 s). Moreover, from Figure 24, one notices that the observer excites the structure after estimation. This explains the higher amplitude and settling time of the control voltage. The excessive augmentation of the observer gain may lead to a control system sensitive to less or non-controlled modes. Moreover, additional decrease of the output noise covariance does not improve either the damping performance or the control voltage.

It is clear, from equation ( 23), that accounting for a state observer increases the system dimension. However, as presented previously, the observer design does not a!ect the control design, unless the modi"cation of the control voltage induced by the observer is accounted for. Hence, the controller and the observer may be designed separately. For the particular case studied in this section, the LQG controller with V"10\ 8 leads to a similar performance with less control voltage than the LQR one. However, it should be noticed that, in practical cases, the output noise covariance is inherent to the signal processing equipment and could not be chosen as a control parameter. Thus, performance as good as for a LQR controller may not always be possible. Moreover, the LQG algorithm did not account for modelling errors, since both observer and controller were based on perfect knowledge of the system dynamics. Alternatively, this could be done using a robust controller as presented in reference [7].

CONCLUSIONS

Using a "nite-element model able to handle sandwich beams with piezoelectric laminated surface layers and viscoelastic core, the design and analysis of the piezoelectric active control of damped sandwich beams has been presented. The piezoelectric direct and converse e!ects were represented through additional electrical d.o.f., condensed at the element level. Lesieutre's anelastic displacement "elds model was used to represent the frequency dependence of the viscoelastic material properties. This led to an augmented state space system, which was then reduced using a complex modal base that eliminates higher frequencies elastic and overdamped relaxation modes. The complex reduced-order system was then transformed to an equivalent state space real representation to allow system analysis and control. Linear quadratic regulator, linear quadratic Gaussian and derivative feedback algorithms were then applied to the reduced-order model. They were all used in an iterative form to account for maximum control voltage, thus guaranteeing control feasibility and preventing piezoelectric material depoling.

Parametric analyses of the open-and closed-loop damping performances of a cantilever damped sandwich beam with bonded piezoelectric actuator and sensor were performed. Actuator length and viscoelastic core thickness were optimized to yield good passive and hybrid damping performances for the "rst three bending modes. Results have shown that optimal performances are obtained for long actuators and thin viscoelastic cores. It was also shown that, in the optimal hybrid con"guration, the LQR algorithm improves the "rst modal damping by 400%, while retaining the passive damping of the remaining modes, even uncontrolled ones. Application of a derivative feedback controller to the sandwich beam optimal geometrical con"guration has led to less e!ective damping as compared to the LQR controller. In fact, both tip de#ection and sensor voltage derivative feedback controllers were not performant to control the "rst bending mode, although the latter was able to control other modes. These results have con"rmed the dependence of this algorithm performance on the actuator and sensor relative localization. However, the well-known spillover destabilizing e!ects of derivative feedback controllers were attenuated by the increase of stability margins provided by the viscoelastic damping. In addition, analysis of the e!ect of an output noise in an LQG full-state optimal observer/controller was performed. It was shown that using the LQG algorithm, the control parameterization, robustness and performance of LQR may be obtained while measuring only the output y, instead of all state variables. Moreover, for the speci"c case studied in this work, the delay e!ect induced by the state estimation of LQG was quite advantageous, since a damping performance similar to that of LQR was obtained with a reduced maximum control voltage.

Other hybrid active}passive damping con"gurations, obtained by several arrangements of piezoelectric and viscoelastic layers, acting conjointly or separately, have been studied and compared but will be presented in a forthcoming paper [START_REF] Trindade | [END_REF]. Also, other promising control algorithms, such as the PPF and independent modal space control (IMSC), are being considered for active}passive damping treatments.

Figure 1 .

 1 Figure 1. Piezoelectric laminated sandwich beam "nite element.

Figure 2 .

 2 Figure 2. Cantilever sandwich beam with bonded piezoelectric actuator and sensor.

Figure 3 .

 3 Figure 3. Elastic modulus G and loss factor of 3M ISD112 viscoelastic material evaluated using three series of ADF parameters (dashed lines) and measured ones (solid lines).

Figure 4 .

 4 Figure 4. Frequency response function between the perturbation force and piezoelectric sensor using the full model (solid lines) and real reduced-order model (dashed lines).

Figure 5 .

 5 Figure 5. Sum of the "rst three open-loop modal damping factors of the sandwich beam.

Figure 6 .

 6 Figure 6. Sum of the "rst three closed-loop modal damping factors of the sandwich beam.

Figure 7 .

 7 Figure 7. Sum of the "rst three damping gains provided by the piezoelectric actuator.

Figure 8 .

 8 Figure 8. Open-and closed-loop FRF of the sandwich beam, measured by the tip de#ection and loaded by the perturbation transverse force, for optimal parameters a"70 mm and h T "0)1 mm: *, passive; ---, hybrid.

Figure 9 .

 9 Figure 9. Open-and closed-loop transient responses of the sandwich beam tip de#ection for optimal parameters a"70 mm and h T "0)1 mm: *, passive; ---, hybrid.

Figure 10 .

 10 Figure 10. Piezoelectric actuator control voltage for optimal parameters a"70 mm and h T "0)1 mm.

Figure 11 .

 11 Figure 11. Root locus between the velocity sensor on the beam tip and the piezoelectric actuator (open-loop poles, ;, and zeros, ᭺, and closed-loop poles, #, for K B "605).

Figure 12 .

 12 Figure 12. Open-and closed-loop FRF of the sandwich beam, measured by the piezoelectric sensor voltage and loaded by the perturbation transverse force, for three velocity feedback control gains. K B values: *, 0; ---, 137;

Figure 13 .

 13 Figure 13. Open-and closed-loop transient responses of the beam tip de#ection for three velocity feedback control gains: K B values: )))))), 0; ---, 137; ) } ) }, 302; *, 605.

Figure 14 .

 14 Figure 14. Control voltages imposed to the piezoelectric actuator for three velocity feedback control gains. K B values: ---, 137; ) } ) }, 302; *, 605.

Figure 15 .

 15 Figure 15. Root locus between the piezoelectric sensor and actuator bonded on the opposite surfaces of the sandwich beam (open-loop poles, ;, and zeros, ᭺, and closed-loop poles, #, for K B "!13)5;10\).

Figure 16 .

 16 Figure 16. Open-and closed-loop FRF of the sandwich beam, measured by the piezoelectric sensor voltage and loaded by the perturbation transverse force, for three voltage rate feedback control gains. K B values: *, 0; ---, !4)5e!3; ) } ) }, !9)0e!3; )))))), !13)5e!3.

Figure 17 .

 17 Figure 17. Open-and closed-loop transient responses of the beam tip de#ection for three voltage rate feedback control gains. K B values: )))))), 0; ---, !4)5e!3; ) } ) }, !9)0e!3; *, !13)5e!3.

Figure 18 .

 18 Figure 18. Control voltages imposed to the piezoelectric actuator for three voltage rate feedback control gains. K B values: -----, !4)5e!3; ) } ) } ) }, !9)0e!3; **, !13)5e!3.

Figure 19 .

 19 Figure 19. Transient response of the sandwich beam controlled by an LQG with output noise covariance V"1: *, passive; ---, hybrid LQR; ) } ) }, hybrid LQG.

Figure 20 .

 20 Figure 20. Control voltage for the LQG controller with output noise covariance V"1: *, LQR; ---, LQG.

Figure 21 .

 21 Figure 21. Transient response of the sandwich beam controlled by a LQG with output noise covariance V"10\: *, passive; ---, hybrid LQR; ) } ) }, hybrid LQG.

Figure 22 .

 22 Figure 22. Control voltage for the LQG controller with output noise covariance V"10\: *, LQR; ---, LQG.

Figure 23 .

 23 Figure 23. Transient response of the sandwich beam controlled by a LQG with output noise covariance V"10\: *, passive; ---, hybrid LQR; ) } ) }, hybrid LQG.

Figure 24 .

 24 Figure 24. Control voltage for the LQG controller with output noise covariance V"10\: *, LQR; ---, LQG.

TABLE 1

 1 Properties of the damped sandwich beam with bonded piezoelectric patches

			Aluminum	PZT5H	ISD112
	Modi"ed Young's modulus (GPa)	79)86 5 )5	various
	Density (kg m\)	2690	7500	1600
	Modi"ed e	constant (C m\)	*	!23)2	*
	Modi"ed e	constant (F m\)	*	1)54;10\	*
	Length, L (mm) Thickness, h @	280	"35) a"50 (d A	280

TABLE 2

 2 Optimal results of hybrid control of damped sandwich beam

		(%)	(%)	(%)	(%)	(%)	t Q	(s)	y max (mm)
	Passive	5)01	13)36	16)17	15)94	15)57	0)60	1)50
	Hybrid	24)85	13)63	16)36	15)94	15.57	0)10	1)07

TABLE 3 Modal

 3

	Gain w R * Gain < Q 1	13)9 16)72 )44 )84 )58 3 )7 !2)6 !7)3 !4)27 )6

damping factors (%) for w R * and < Q 1 feedback at maximum gain 1234 5 Passive 5)21 3 )41 6 )31 6 )01 5 )6

ACKNOWLEDGMENTS

This research was supported by the &&DeH leH gation GeH neH rale pour l'Armement'', Advanced Materials Branch, under contract D.G.A./D.S.P./S.T.T.C./MA. 97-2530, which is gratefully acknowledged.