Servane Gillet

Jean-François Ponge
email: jean-francois.ponge@wanadoo.fr

Species assemblages and diets of Collembola in the organic matter accumulated over an old tar deposit

Keywords: Collembolan communities, Gut contents, Polycyclic Aromatic Hydrocarbons 1. Introduction

An oil refinery was abandoned in 1964 at Merkwiller-Pechelbronn (Alsace, France). We investigated the food diets and species assemblages of Collembolan communities living in a thick pasty tar deposit overlaid by an organic soil, by comparison with an adjacent unpolluted plot. At the polluted plot the species richness of the springtail community was low (9 species, versus 27 at the unpolluted plot) but its total abundance was at the same level in both plots. Most springtails consumed more fungi than bacteria at the polluted plot, contrary to the unpolluted plot. Changes in habitat features and food resources might explain the observed decrease in local biodiversity, rather than direct PAH toxicity.

The direct toxicity of pollutants to soil animals has been widely reported and indirect effects, for instance through changes in habitat and food quality, are probable [START_REF] Blakely | Soil invertebrate and microbial communities and decomposition as indicators of polycyclic aromatic hydrocarbon contamination[END_REF][START_REF] Gillet | Changes in species assemblages and diets of Collembola along a gradient of metal pollution[END_REF].

Polycyclic Aromatic Hydrocarbons (PAHs), such as naphthalene, phenanthrene, or benzo[a]pyrene, are toxic aromatic compounds [START_REF] Bispo | Toxicity and genotoxicity of industrial soils polluted by polycyclic aromatic hydrocarbons (PAHs)[END_REF][START_REF] Schirmer | Ability of 16 priority PAHs to be directly cytotoxic to a cell line from the rainbow trout gill[END_REF] which are nonpolar, hydrophobic and neutral molecules composed of two or more fused benzene rings in linear, step, or cluster arrangement. Nowadays, their presence in the soil results mainly from industrial activities such as coke production, petroleum refining and other hightemperature industrial processes, which induce pollution by accidental oil spilling and waste deposition, even long after site abandonment. Field studies revealed that the presence of PAHs decreases the biodiversity of soil animal communities and changes their vertical distribution [START_REF] Best | Response of microarthropod populations to naphthalene in three contrasting habitats[END_REF][START_REF] Erstfeld | Effects of chronic low-level PAH concentration on soil invertebrate communities[END_REF]. Laboratory bio-assays have demonstrated toxic and anaesthetic effects of PAHs on soil animals [START_REF] Crouau | The use of Folsomia candida (Collembola, Isotomidae) for the bioassay of xenobiotic substances and soil pollutants[END_REF][START_REF] Chenon | Genetic polymorphism in nine clones of a parthenogenetic collembolan used in ecotoxicological testing[END_REF][START_REF] Dorn | Assessment of the acute toxicity of crude oils in soils using earthworms, Microtox®, and plants[END_REF], as well as bioaccumulation [START_REF] Johnson | Temporal changes in earthworm availability and extractability of polycyclic aromatic hydrocarbons in soil[END_REF] and avoidance [START_REF] Michelozzi | Eco-ethological analysis of the interaction between isoprene and the behaviour of Collembola[END_REF].

Our study site was a petroleum refinery abandoned in 1964. It still displays a patchy pollution of soil and ponds by PAHs. The aim of our work was to study the impact of this type of pollution on soil fauna by a comparison between a plot polluted by PAHs and a nearby unpolluted control plot. In a parallel study on the composition of humus profiles, we observed that the organic matter which accumulated over the 40year-old tar deposits was permeated by a large population of mycorrhizal fungi, in contrast to the adjacent unpolluted plot which was characterized by the rapid incorporation of litter to mineral matter through earthworm activity [START_REF] Gillet | An optical analysis of the organic soil over an old petroleum tar deposit[END_REF]. We used a community-level approach to evaluate the impact of PAHs on the saprophagous compartment of the ecosystem. The collembolan community (Arthropoda: Hexapoda) was chosen for its great abundance and diversity in a wide variety of soil types [START_REF] Petersen | A review of collembolan ecology in ecosystem context[END_REF][START_REF] Ponge | Biocenoses of Collembola in atlantic temperate grass-woodland ecosystems[END_REF] and the flexibility of its alimentary habits [START_REF] Gillet | Changes in species assemblages and diets of Collembola along a gradient of metal pollution[END_REF][START_REF] Hasegawa | Changes in feeding attributes of four collembolan populations during the decomposition process of pine needles[END_REF][START_REF] Ponge | Vertical distribution of Collembola (Hexapoda) and their food resources in organic horizons of beech forests[END_REF]. Population indices and food bolus -3analyses were used to follow changes in function and biodiversity of collembolan populations under the influence of PAH pollution.

Materials and methods

Study site

The study was conducted at the site of the petroleum refinery of Merkwiller-Pechelbronn, about 50 km north of Strasbourg (Alsace, France). The refinery had an intensive activity until 1964, afterwards it was progressively dismantled then totally abandoned. The site is located on the Merkwiller-Pechelbronn oil field (bituminous sand) on the western edge of the Rhine rift valley. The superficial soil (5 m) is composed of recent fluvial deposits overlaying 1400 m thick sediments above the granitic substratum [START_REF] Sittler | Les hydrocarbures d'Alsace dans le contexte historique et géodynamique du fossé Rhénan[END_REF].

Nowadays, the site area (20 ha) is characterised by the presence of old buildings now included in a great variety of naturally established ecosystems (woodland, grassland, ponds) with zones polluted by hydrocarbons, in particular by tar deposits. Two plots (polluted versus unpolluted used as control), were sampled in October 2002. They were considered to be representative of the patchy occurrence of tar, as ascertained by the examination of a number of adjacent (polluted and unpolluted) soil profiles. The polluted plot, approximately 20 m 2 , was characterised by a thick pasty tar patch overlaid by a 3-5 cm organic layer, a discontinuous field layer composed of Hedera helix, Geranium robertianum, Carex pilosa, Solidago canadensis and Taraxacum officinale, a shrub layer composed of Fraxinus excelsior, Rubus fruticosus and Salix capraea and a tree layer composed of Acer campestre, Betula pendula and Quercus robur. The unpolluted, control plot was located 15 m south of the polluted plot, and its surface was arbitrarily fixed to 20 m 2 , for the sake of

Sampling and identification of animals

Collembola were sampled on the organic matter accumulated over the tar deposit and in the adjacent unpolluted plot. At each plot, five soil cores, 5 cm diameter and 10 cm depth (litter included), were collected with a core sampler and, after discarding tar pieces, stored in plastic bags. Springtails were extracted from soil/litter by the dry-funnel method [START_REF] Edwards | A comparison of extraction methods for terrestrial arthropods[END_REF] and preserved in 95% (v/v) ethyl alcohol until sorting and identification. The animals were sorted under a dissecting microscope, then mounted in chloral-lactophenol (25 ml lactic acid, 50 g chloral hydrate, 25 ml phenol). They were identified to species level using a phase contrast microscope at 500x and several keys and diagnoses [START_REF] Bretfeld | Synopses on Palaearctic Collembola. II. Symphypleona[END_REF][START_REF] Christiansen | A catalogue of the species of the genus Pseudosinella[END_REF][START_REF] Gisin | Collembolenfauna Europas[END_REF][START_REF] Hopkin | A Key to the Springtails of Britain and Ireland, AIDGAP Test Version[END_REF][START_REF] Potapov | Synopses on Palaearctic Collembola. III. Isotomidae[END_REF][START_REF] Stach | The Apterygotan Fauna of Poland in Relation to the World-Fauna of this Group of Insects. Tribe: Orchesellini[END_REF][START_REF] Stach | The Apterygotan Fauna of Poland in Relation to the World-Fauna of this Group of Insects. Tribe : Entomobryini[END_REF][START_REF] Zimdars | Synopses on Palaearctic Collembola. Part I. Tullbergiinae[END_REF]. We calculated in each sample the total abundance, the the species richness, the Shannon Index and the equitability. We also calculated the total species richness of the five samples taken at each plot.

Observation of gut contents

Gut contents from each mounted Collembola were observed under phase contrast at 500x magnification and classified in twelve categories:

 Melanized hyphae  Hyaline hyphae  Fungal spores  Bacteria  Mineral particles
 Holorganic humus (amorphous organic matter)  Hemorganic humus (mixture of organic matter and mineral particles)

 Micro-algae  Pollen grains  Empty guts
The proportion in volume of the different categories was visually estimated to the next 5% in each animal gut then averaged by species and by site.

Chemical analyses

Samples were collected in April 2003 to determine the amount of PAHs from the EPA list [START_REF] Greene | Protocols for Short-Term Toxicity Screening of Hazardous Waste Sites[END_REF] in (1) the 10 top cm of the A horizon of the control soil (2) the tar deposit at 10-20 cm depth and (3) the OF horizon overlying the tar deposit. To check the validity of our control plot, samples were collected in May 2003 in the park of the laboratory (Brunoy, Ile-de-France, France), in the 10 top cm of a similar soil (rich earthworm mull at neutral pH) under deciduous woody vegetation. Composite samples were taken from each plot, then kept in glass jars and rapidly transported to the laboratory. Then, soil samples were homogenised and sieved at 1 cm then kept in glass jars at -18C° until analysis. Each sample was defrozen, dried, then sieved at 2 mm.

PAHs were extracted from each sample with the automatic system ASE 200 (Accelerated Solvent Extraction) DIONEX, using a mixture of dichloromethane and acetone (50/50) for soil and acetonitrile for tar. The extract was concentrated under forced air with a TuroVap LV (Zymark), then PAHs were separated by HPLC (High Power Liquid Chromatography) with UV detection (alliance 2690 chain, PDA 996 detector, column LC-PAH Supelco). After sieving the contaminated soil, which was mainly made of badly decomposed tree litter, only a small amount of fine matter was available for analysis, and only 7 major PAHs could be analysed on this material: naphthalene, phenanthrene, fluoranthrene, pyrene, benzo(a)anthracene, chrysene and benzo(ghi)perylene, the others being under detection level (0.02 µg/g for uncontaminated soil, 0.04 µg/g for contaminated soil, 0.1 µg/g for tar). All measurements were done in triplicate.

After extraction of fauna, air-dried soil was stored in plastic bags for pH measurement. The soil was suspended in deionized H 2 0 and 1M KCl (1:5 soil:water v:v) for pH H 2 0 and pH KCl, respectively. Each suspension was shaken for five minutes, then pH was measured in the supernatant after sedimentation for 2 h with a glass electrode. The difference between these two measurements (pH) was taken as an estimate of exchange acidity.

Statistical treatment

Non-parametric Mann-Whitney tests were performed to compare chemical parameters (pH) and population parameters (abundance, species richness, diversity and equitability of collembolan communities) between sites, using samples as replicates (five in each site).

Results

Chemical analyses

The distribution of the 16 PAHs of the EPA list and the total amount of PAHs were quite similar in the control soil from the Merkwiller-Pechelbronn site and in the soil from the park of the laboratory (Table 1). As a consequence we estimated that the plot chosen as a control at Merkwiller-Pechelbronn was valid. The total amount of the seven PAHs analysed in the soil over the tar deposit at Merkwiller-Pechelbronn (5.95 µg/g) was six times higher than the corresponding amount at the control plot (0.93 µg/g). The concentration of benzo(ghi)perylene was eleven times higher than the control, while the amount of fluoranthrene was only three times higher.

The pH measured in water and potassium chloride was neutral and did not differ between both sites at Merkwiller-Pechelbronn (Table 1). However, mann-Whitney U test revealed that exchange acidity, expressed by pH, was higher on the unpolluted site.

Collembolan communities

At the unpolluted plot 27 species were found whereas only 9 were found in the polluted zone. The species richness per sample, the Shannon Index and the equitability were higher at the control than at the contaminated plot (Table 2). However, the total abundance of springtails did not differ between plots.

Only five species were common to both polluted and unpolluted plots: Folsomia candida, Parisotoma notabilis, Sminthurides malmgreni, Sminthurides parvulus and Sminthurinus aureus. Among these species, S. aureus was significantly more abundant in the control soil (x 4). Over the tar deposit, 83% of the total abundance was represented by Proisotoma minuta (43%) and Parisotoma notabilis (40%). P. minuta was present in all samples taken at the polluted plot while it was absent from all samples taken at the control plot. The abundance of this species (together with that of Paristoma notabilis in most samples from the tar deposit) explains why the total abundance of Collembola did not differ between plots in spite of a much lower number of species at the polluted plot (Table 2).

Diets of Collembola

The food diet of the three species Parisotoma notabilis, Folsomia candida and Sminthurinus aureus was compared between plots (Figure 1). Fungal mycelium was more common at the polluted plot, while the contrary was observed for bacteria, except in F. candida guts which showed bacteria in equal proportion in both sites but more hyaline hyphae and hemorganic humus over the tar deposit. The proportion of empty guts was higher at the polluted plot for P. notabilis and S. aureus.

The diet of S. aureus was more diverse at the unpolluted plot, containing microalgae, pollen, humus, bacteria and fungal spores and mycelia, but only fungal mycelium in the polluted plot. However the latter observation could be an artifact due to the low number of specimens (3) collected over the tar deposit. The gut of the bacteriophagous species M. minimus contained 68% bacteria, and this species was only present at the unpolluted site, where it was found in all samples. The isotomid P. minuta, which was present only at the polluted site where it dominated the community, showed a high proportion of empty guts (67%), 21% mycelium and few bacteria (2%).

Discussion

In spite of a slight distance between plots and a low concentration of PAHs over the tar deposit (Table 1), a great dissimilarity in collembolan communities was observed between the organic soil accumulated over the tar deposit and the adjacent control soil (Table 2). It should be noticed that sieving of the soil to 2 mm before PAH extraction (see Materials and methods) could have led to underestimate PAH concentration in the soil over tar. The toxicity of crude oil for Collembola was shown by Sendstad [START_REF] Sendstad | Accelerated biodegradation of crude oil on arctic shorelines[END_REF] who found 2% of total Collembola in a polluted plot compared to a control plot. Three years after application, toxicity was still present given the absence of recolonisation by fauna of the polluted zone. In our study, the collembolan community could be affected by i) the presence of pollutants in the soil, through direct effects of PAHs or other toxic compounds on the most sensitive species and/or ii) indirect effects such as changes in food resources. Like in a previous study on a site polluted by heavy metals [START_REF] Gillet | Changes in species assemblages and diets of Collembola along a gradient of metal pollution[END_REF], the species richness of springtail communities was observed to decrease under the influence of pollution, at constant or even higher total abundance. In both cases, this could be explained by an increase in free habitats liberated by most sensitive species. Isotomids seemed to be less sensitive to PAH pollution, as was also observed by Blakely et al. [START_REF] Blakely | Soil invertebrate and microbial communities and decomposition as indicators of polycyclic aromatic hydrocarbon contamination[END_REF]. Here this group was represented by the two dominant species Proisotoma minuta and Parisotoma notabilis.

In our study, changes were observed in the diet of springtails under the influence of pollution by hydrocarbons. This shift can be explained by a change in the use of food resources, since fungi were favoured to the detriment of bacteria. This possibility reflected a change in the composition of microbial communities in the favour of fungi, which could explain why the bacteriophagous Megalothorax minimus was only present at the unpolluted plot. Micromorphological analyses showed a high amount of ectomycorrizal fungal hyphae in the organic matter accumulated over the tar deposit with a poor earthworm activity, while a rapid incorporation of organic matter to mineral matter through earthworm activity was registered in the control soil [START_REF] Gillet | An optical analysis of the organic soil over an old petroleum tar deposit[END_REF]. Shifts in microbial communities in favour of fungi could be better explained by a change in humus form (and associated foodwebs) than by the direct toxicity of PAHs to microorganisms [START_REF] Blakely | Soil invertebrate and microbial communities and decomposition as indicators of polycyclic aromatic hydrocarbon contamination[END_REF].

We found strong changes in invertebrate communities and alimentary habits of soil animals forty years after abandonment of industrial activity. Despite a low PAH concentration in the soil accumulated over the tar deposit (Table 1), the inability of earthworm populations to colonize pollution spots, and the development of a superficial ecto-mycorrhizal root mat caused significant changes in the environment of animal as well as microbial communities. Given the shift from mull (dominated by earthworm activity) to moder (dominated by enchytraeid activity) ascertained by micromorphological analysis [START_REF] Gillet | An optical analysis of the organic soil over an old petroleum tar deposit[END_REF], the observed decrease in biodiversity and change in the food diet of the Collembolan community can be ascribed, at least partly, to a change in humus form [START_REF] Ponge | Humus forms in terrestrial ecosystems: a framework to biodiversity[END_REF]. Equitability 0.8 0.9 0.9 1.0 0.9 0.7 0.9 0.8 0.4 0.7 *

Control soil Soil over tar

Collembolan densities per sample (20 cm 2 ) at unpolluted and polluted plots. Significant differences betw een groups (Mann-Whitney U test, P = 0.05) are indicated by stars

Figure legends Fig 1 .

 legends1 Figure legends

  Fig. 1 2

  comparison, although it did not differ from the rest of the site. It was characterised by a much greater plant biodiversity and a good earthworm mull humus. The field layer was

	composed of Hedera helix, Arum maculatum, Carex pilosa, Fragaria vesca, Geranium
	robertianum, Geum urbanum, Potentilla reptans, Stachys sylvatica, Solidago
	canadensis and Taraxacum officinale, the shrub layer was composed of Acer
	pseudoplatanus, Carpinus betulus, Cornus mas, Cornus sanguinea, Crataegus
	monogyna, Fraxinus excelsior, Ligustrum vulgare, Prunus avium, Rosa canina and
	Rubus fruticosus, and the tree layer was composed of Acer campestre, Prunus avium
	and Quercus robur.

Table 1 Tar

 1 Chemical features of tar, organic matter accumulated over tar, control soil from Pechelbronn site and laboratory soil. Concentration of PAH (Polycyclic Aromatic Hydrocarbons) is expressed in μg/g dry soil. Means of three replicated measures on pooled samples (PAHs) or of five replicate samples (pHs) are follow ed by standard errors. Significant differences among groups at 0.05 level (for pH only) are indicated by different superscript letters (Mann-Whitney test). d.l. = detection limit

	Soil over tar	Control soil	Laboratory soil

Table 2 Mann-Whitney

 2

	Dicyrtoma fusca (Lucas, 1842

Acknowledgements

We thank the Agence de l'Environnement et de la Maîtrise de l'Energie (ADEME) for financial support and Vasilica HAMMADE and Laurence BELKESSAM from the Centre National de Recherches sur les Sites et Sols Pollués (CNRSSP, Douai, France) for PAH analyses.


Stefano Vidoli

Francesco Dell'isola

Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks

 were found. These papers deal with piezoelectric continuously distributed actuators embedded in plates and exhibit very interesting and mathematically rigorous homogenization results. It has to be underlined, however, that the device modelled there is different from what is considered here. Indeed, in this paper, the uniformly distributed actuators are interconnected by a resonant passive electrical network.

Introduction

In (dell'Isola and Vidoli, 1998-a) and (dell'Isola and Vidoli, 1998-b) the problem of controlling a certain class of truss modular beams has been addressed; it is shown that the available piezoelectric actuators are in principle able to dampen the mechanical vibrations of the beams. In [START_REF] Vidoli | Modal coupling in one-dimensional electro-mechanical structured continua[END_REF] the concept of continuously distributed control has been introduced and developed for one-dimensional beams. This control is obtained by (i) embedding the actuators in the structural members, (ii) interconnecting them by an electric transmission line.

In the present paper it is proposed to control similarly the vibrations in plates by means of a set of uniformlydistributed electrically-interconnected actuators. However, because the plate is a two-dimensional structural member, such an interconnection must be obtained by means of an electrical two-dimensional uniformlydistributed network. The mathematical diff culty (in comparison with [START_REF] Vidoli | Modal coupling in one-dimensional electro-mechanical structured continua[END_REF]) to be confronted concerns the need of changing the kinematical descriptors of the electric state of the system. This diff culty goes along with the need to conceive a way, which is intrinsically two-dimensional, to suitably interconnect the actuators. Both these diff culties are resolved by: (i) introducing the f eld of e-state (the material time derivative of which is the electric potential); and (ii) conceiving a truly bidimensional circuital interconnection scheme among the actuators which is governed by the same Partial Differential Equation (PDE) valid for a membrane.

We utilize the classical idea of using actuators for achieving damping. However, to the best of our knowledge, nowhere in the literature 1 has a system constituted by: (i) a plate; (ii) an embedded uniformly-distributed set of actuators; (iii) a transmission bidimensional electric network interconneting the actuators, been considered. In the following sections we show that it is possible to f nd a particular circuital topology which allows for eff cient damping of mechanical vibrations. This damping effect is obtained using inductances lower than those needed in the shunt devices proposed up till now (see [START_REF] Guran | Wave Motion, Intelligent Structures and Nonlinear Mechanics[END_REF]). This engineering concept is novel and may be useful in many applications.

Advantages of proposed uniformly distributed net-control systems

The common features of the control devices already conceived are the differentiation between the sensing and the actuation systems and the localization of PZT actuators in a small number of sites of the vibrating structure. Both limit control eff ciency; indeed, the f rst feature implies the need of a coordinating active system, controlling the actions of the actuators in response to the inputs from the sensors. On the other hand the latter feature implies an optimal localization problem -for both actuators and sensors -the solutions of which depend on the particular mechanical vibration mode to be considered (see [START_REF] Fuller | Active Control of Vibration[END_REF]). Moreover it is diff cult to optimize the characteristics of the control system to obtain low equivalent impedances -these are required to allow for a relevant energy transformation from the mechanical to the electrical form -and eff ciently drive the PZT actuators (see [START_REF] Hagood | Damping of structural vibrations with piezoelectric materials and passive electrical networks[END_REF] and [START_REF] Bondoux | Piezo-damping of light structures: modelling and experimental results[END_REF]).

Some efforts to overcome the f rst of these drawbacks have been made. In particular, the concept of selfsensing actuators has been introduced (see [START_REF] Fuller | Active Control of Vibration[END_REF]); an ad hoc electric circuit is connected to the piezoelectric patch allowing its two-fold behavior. However every patch remains isolated and its electromechanical action has to be coordinated with the rest of the structure. Furthermore even when a large number of actuators is used to control the shape of plates (see [START_REF] Batra | Deflectio control during dynamic deformations of a rectangular plate using piezoceramic elements[END_REF]), interconnecting the actuators via a circuital network was never considered.

In the present paper exploiting the concept of 'analogy' between mechanical structures and electric control systems, it is proposed to control a plate-like structure by means of a distribution of actuators connected to an electric transmission network. An internal resonance phenomenon, between structural modes and electric modes, is induced to obtain the maximum control eff ciency.

The net-control system has two practical advantages: ceteris paribus, it requires lower performances to the PZT actuators and allows for: (i) a more eff cient control; and (ii) a reduced time to transfer the energy between electrical and mechanical forms. Moreover the net-control system bypasses the problems of optimal positioning (of actuators and sensors) being able to manage all mechanical modes thought the same distributed conf guration of its collocated actuators.

In order to prove the previous statements a mathematical model for the conceived apparatus is used to study its coupled electro-mechanical vibrations. Using numerical simulations for square plates, results of a more general nature are obtained. Indeed the considered case yields important information regarding how to establish the coupling of electrical with mechanical modes. The very nature of the tuning of the electrical network to the mechanical propagation phenomena is local and therefore independent of the global geometry of the structural member.

After choosing a circuital topology, tuning is obtained by suitably determining circuital impedences. This is done by means of standard mathematical tools, however, the electro-mechanical coupling effect under investigation is not studied nor is the energy exchange in inf nite-dimensional partitioned dynamical systems explicitly considered in the literature (for a detailed discussion of this point see [START_REF] Vidoli | Modal coupling in one-dimensional electro-mechanical structured continua[END_REF]). Therefore we needed to develop an electro-mechanical coupling criterion and f nd an optimality condition for mechanical energy dissipation.

For dealing with more complicated geometries, suitable f nite element schemes will be needed. However, the simple case presented herein seems to be of both theoretical and practical interest. It will allow for a better understanding of the general properties of the conceived device and therefore will direct the design of piezoelectromechanical plates with more complex geometries and will represent a benchmark for future numerical simulation.

Description of the system

In this paper it is proposed to control the mechanical vibrations of a plate with a distribution of piezoelectric actuators interconnected by means of electric impedances, as shown in figu e 1: gray boxes represent the actuators while the black ones represent the electric impedances. The concept of electrical interconnection of piezoelectric actuators represents the main feature of the considered device. It has to be stressed that this interconnection can be obtained by means of several different circuital topologies; one possible topology is reproduced in figu e 2. 2 The fundamental topological difference between the considered bidimensional network and a transmission line has to be underlined: a node is grounded by means of an actuation device and is connected to four other nodes. This circumstance is accounted for mathematically by introducing a PDE governing the evolution of the electric state descriptor.

From an electrical viewpoint the actuators can be considered as capacitors: therefore they are able to store electric energy in DC regimes. A capacitor is a one-port circuital element with two terminals. In the connection scheme represented in figu e 2 one of these terminals is connected to the electric network while the other is grounded. On the other hand, from a mechanical viewpoint, the actuators: (i) exert mechanical actions in response to electric inputs; (ii) contribute to the structural stiffness of the structure to which they are mechanically connected; and (iii) provide electric signals in response to mechanical deformations.

Therefore the piezoelectric actuators establish an electro-mechanical interaction and allow for the transformation of mechanical into electrical energy. On the other hand the interconnecting electric network allows the electric signal and electric energy to travel simultaneously with the mechanical waves.

The device described in this subsection will be called a piezo-electromechanical plate.

Mathematical model

To start the study of the behavior of a piezo-electromechanical plate, a continuum model is introduced which allows for an accurate description of vibrations when the involved wave-lengths are not too small compared to the dimensions of the single actuator.

The procedure which is followed leads to a homogenized, continuous model that is in some way similar to the homogenized model introduced in [START_REF] Hoffmann | Homogenization of a model describing vibration of nonlinear thin plates excited by piezopatches[END_REF][START_REF] Botkin | Homogenization of an equation describing linear thin plates excited by piezopatches[END_REF] and [START_REF] Hoffmann | Homogenization of von Kármán plates excited by piezoelectric patches[END_REF]. However we limit ourselves to heuristically deduce it with a formal identif cation in powers, as done usually in the theory of plates (see the classical results found in [START_REF] Reissner | Selected Works in Applied Mechanics and Mathematics[END_REF]) or in a different context e.g. by [START_REF] Di Carlo | Continuum modelling of a beam-like latticed truss: identif cation of the constitutive functions for the contact and inertial actions[END_REF]dell'Isola et al., 1998). A rigorous mathematical proof -establishing that the homogenized model we f nd is really a -limit of the micro-model from which we are starting -is not attempted.

The subsequently postulated balance of power allows us to obtain a set of PDEs for the f elds describing the electric and mechanical states.

In the following formulation, the reader will f nd a multiplicity of mathematical symbols; for instance one will need to distinguish between the Laplacian (respectively gradient) operator in three dimensions and the Laplacian (respectively gradient) operator in two dimensions. This need is justif ed when one considers that the aim is to deduce some equations for a 2-D plate starting from the three-dimensional equations of a Cauchy 3-D continuum. This multiplicity seems to be unavoidable, as it ref ects the complexity of the physical system we try to describe.

We follow the standard mathematical notation; for instance given a second-order tensor A we will denote its symmetric part by sym A and its trace by tr A. Given a two-dimensional surface S embedded in the threedimensional Euclidean space we will denote by ∂S its boundary in the sense of Gaussian differential geometry, × will denote set Cartesian product, • will denote the inner product between vectors, an upper dot will denote the time derivative of considered f elds. Finally ⊗ will denote the standard tensor product operation.

Equation of motion

We deal with a plate body B occupying the region C = S × I, where S is a plane surface and I the real interval [-h, h]. As usual, the thickness 2h is assumed to be small compared with the diameter of S.

Moreover we assume that the following balance of power:3

S I (b • u) + ∂S I (f • u) + S (i ψ) + ∂S (χ ψ) = S I S • (sym Grad u) + S (I • grad ψ), (1)
must hold for every test f eld ( u, ψ). Here u represents the displacement f eld, while ψ is the electric potential f eld. As a consequence u is the velocity f eld and ψ is the time-integral of the electric potential difference between the nodes and the ground. Moreover b and f, i and χ , Grad and grad are the body and surface external forces, the body and surface current densities from the ground, and the gradient operators in C and S respectively. Finally S is the stress tensor and I is the current vector in the network.

According to the geometry of the body, the position vector is decomposed as:

x = r + ζ e, ( 2
)
where r is the position vector in S, ζ ∈ I and e is the unit vector perpendicular to S. To deduce from the 3-dimensional Cauchy model of B the behavior of a bending plate, we use the Kirchhoff-Love compatible identif cation procedure based on the following kinematical reduction map for displacements:

u(r, ζ ) = w(r)e-ζ grad w(r). ( 3
)
As it is easily checked, the previous equation can be interpreted mechanically stating that every material segment orthogonal to the mean surface of the plate in the reference conf guration remains: (i) orthogonal in every considered current conf guration; and (ii) has a constant length. The function w models the transverse displacement of the material particles of the plate. It is assumed that the electric potential ψ depends on r only.

It follows that the inf nitesimal deformation f eld is expressed as (grad grad denotes the Hessian differential operator):

E = sym(Grad u) = -ζ sym(grad grad w). (4)
Substituting in the equation ( 1) the reduction map (3) we obtain:

S (b ẇ + B • grad ẇ + i ψ) + ∂S (T ẇ + m• grad ẇ + χ ψ) = S [M • sym(grad grad ẇ) + I • grad ψ], (5)
where

M = - I ζ S| S , B = - I ζ b| S , b = I b • e, (6)
T = I (f • e), m = I -ζ f| S , (7)
are the dynamical generalized forces in the reduced plate model. Applying the divergence theorem (div denotes the divergence operator applied to the following tensor and ν is the outward pointing normal vector to ∂S belonging to the tangent plane to S), we get:

S (b -div B -div div M) ẇ + (i + div I) ψ + ∂S T + (B + div M) • ν ẇ + (m -Mν) • grad ẇ + (χ -I • ν) ψ = 0 (8)
which must hold for every admissible velocity f eld ( ẇ, ψ); this condition leads to the following balance equations:

div div M + div B -b = 0, div I + i = 0, on S, (9)
and to the following reduced expression of ( 8):

∂S T + (B + div M) • ν ẇ + (m -Mν) • grad ẇ + (χ -I • ν) ψ = 0 (10)
from which we can get well-posed boundary conditions. The equations ( 9) and ( 10) represent a reduced model for the piezo-electromechanical plate.

Assuming that the body B is linear, isotropic and homogeneous (so that we do not need to distinguish in the power balance the actual from the reference conf guration) and that the network is linear and dissipative, we get:

S = 2µ L E + λ L (tr E)id, b = -ρ ü, -grad ψ = L N İ + R N I, (11)
where µ L and λ L are the Lamè moduli, ρ the mass density, L N and R N are respectively the net-inductance and net-resistance and id is the identity operator in C. As a consequence the part of the reduced constitutive equations which does not depend on piezoelectro-mechanical coupling (which we denote M m and i e ) reads as follows:

M m = J I 2µ L sym (grad grad w) + λ L (lapl w)id , i e = k C d 2 ψ, (12)
where J I = 2h 3 /3, lapl is the Laplacian operator in S and k C is the purely electric grounded capacitance, while d 2 represents the area of inf uence of the actuator (namely the area of the plate divided by the number of actuators). For the inertial terms from ( 11) and ( 6) we get:

B = -J I ρ grad ẅ, b = -2hρ ẅ. ( 13
)
However the piezoelectric actuators have a two-fold behavior: from a mechanical viewpoint they enhance the bending stiffness of the plate and produce bending moments in response to applied voltage; from an electric viewpoint they enhance the grounded capacitance per unit area of the electric net and produce a stored capacitive charge in response to applied curvatures.

Let us introduce an orthonormal coordinate system (o, e 1 , e 2 ) in S wherein the partial derivatives of the f eld f with respect to the corresponding coordinate variables x i will be denoted f ,i . Concerning the part of bending moment tensor induced by the piezoelectric effect M e we assume that its component expending power on the component w ,12 of the curvature vanishes so that the following representation holds:

M e = M 11 (e 1 ⊗ e 1 ) + M 22 (e 2 ⊗ e 2 ).
(14)

In addition we specify the properties of the single PZT actuator used by

       M 11 M 22 Q/d 2        =     g mm 0 -g me 0 g mm -g me g me g me g ee            w ,11 w ,22 ψ        , ( 15
)
where M ii and w ,ii are the piezo-electrically induced bending moments and curvatures, while Q/d 2 and ψ are the charge per unit area and voltage between the actuator plates. The constitutive equation ( 15) establishes that the actuators can supply electrically induced moments only in two f xed material orthogonal directions and that they cannot supply 'mixed' moments; the orthogonal system introduced accounts for this directionality. Moreover we have assumed that the piezoelectric stiffnesses in e 1 and e 2 are equal; this assumption seems reasonable when using PZT actuators exploiting Poisson effect.

Therefore the overall constitutive relations for M and i read as follows:

M = M m + g mm [w ,11 (e 1 ⊗ e 1 ) + w ,22 (e 2 ⊗ e 2 )] -g me ψid, (16)
i = i e + Q d 2 = i e + g ee ψ + g me lapl ẇ. ( 17
)
Let us now introduce the bending stiffness and capacitance per unit area of the plate:

D P = J I (2µ L + λ L ), C N := g ee + k C d 2 =: k ee + k C d 2 ; (18)
the balance equations in terms of kinematical f elds become:

D P dlapl w + g mm (w ,1111 + w ,2222 ) -J I ρ lapl ẅ + 2hρ ẅ -g me lapl ψ = 0, -lapl ψ + L N C N ψ + R N C N ψ + R N g me lapl w + L N g me lapl ẇ = 0, ( 19
)
where dlapl denotes the double Laplacian operators in S.

In order to f nd the dimensionless form of (19) we introduce as spatial characteristic length the diameter of the plate , and def ne v := w/ , φ := ψ/ V , the characteristic pulsation4 ω = π D P M P -being M P = 2ρ 2 h the total mass of the plate -so that:

D P 2hρ 4 ω 2 v + g mm 2hρ 4 ω 2 (v ,1111 + v ,2222 ) - h 2 3 2 v + v - g me V M P ω φ = 0, - 1 L N C N 2 ω 2 φ + φ + R N L N ω φ + g me C N ω V v + R N g me L N C N V ω 2 v = 0; (20)
where is the dimensionless Laplacian operator, ( ) ,i and the upper dot denote respectively the dimensionless space and time derivative. Let us introduce the following Sobolev norm:

f H 2 = f 2 + i (f ) 2 ,i + ij (f ) 2 ,ij ; (21)
then we will assume that:

g mm 2hρ 4 ω 2 D P 2hρ 4 ω 2 , h 2 3 2 v H 2 v H 2 . ( 22
)
This last inequalities are valid when the wave-length is much bigger than the plate thickness and when one considers only the lower spatial eigenmodes.

The characteristic e-state parameter V := √ M P /C N is chosen to maintain the symmetry, so that (20) becomes:

α v + v -γ φ = 0, -β φ + φ + γ v + δ φ + δγ v = 0, (23)
where

α := D P M P 2 ω 2 = 1 π 2 , β:= 1 L N C N 2 ω 2 , ( 24
) γ := g me ω 1 M P C N , δ:= R N L N ω (25)
are dimensionless numbers. We remark that when the electro-mechanical coupling parameter γ vanishes, ( 23) reduces to the uncoupled system of the Kirchhoff-Love plate and membrane-like electric network equations:

α v + v = 0, φ + δ φ = β φ.
(26)

Partitioned modal analysis

In this section we adapt to the set of equations ( 23) the reasoning developed in [START_REF] Vidoli | Modal coupling in one-dimensional electro-mechanical structured continua[END_REF].

Let H m and H e be the subspaces of L 2 (S), the space of R-valued square-integrable functions def ned on S verifying suitable homogeneous boundary and smoothness conditions. Let v, v, v ∈ H m and φ, φ, φ ∈ H e . Let L mm and L ee be linear self-adjoint differential operators on H m and H e respectively, and G A me indicate the adjoint of the linear differential operator G me from H e to H m . We consider the following evolutionary problem:

αL mm (v) + v -γ G me ( φ) = 0, βL ee (φ) + φ + γ G A me ( v) + δ φ + δγ G A me (v) = 0, (27)
starting from suitable initial conditions for v and φ. We remark that equations ( 23) have the structure of ( 27).

The subscripts m and e stand for mechanical and electrical respectively.

In order to study the interaction between the electrical and mechanical components of state descriptors we introduce in H m and H e the eigenbases supplied by the spectral representations theorem for the self-adjoint operators L mm and L ee respectively. Therefore, for every v ∈ H m , φ ∈ H e , we have [START_REF] Reed | Methods of Modern Mathematical Physics[END_REF]:

v = h v h m h , φ= k φ k e k , ( 28
)
L mm (v) = h λ h v h m h , L ee (φ) = k ν k φ k e k . ( 29
)
Here λ h and ν k respectively denote the eigenvalues of L mm and L ee , m h and e k are the corresponding eigenfunctions, v h := v, m h L 2 and φ k := φ, e k L 2 are the time-dependent Fourier coeff cients. If we def ne the scalars:

C hk := m h , G me (e k ) L 2 , ( 30
)
C kh := e k , G A me (m h ) L 2 = G me (e k ), m h L 2 = C hk , ( 31
)
and consider that m h and e k are bases of H m and H e as eigenfunctions of self-adjoint operators, equations ( 27) can be written:

vh + αλ h v h -γ k C hk φk = 0, φh + βν h φ h + δ φh + γ k C kh (δv k + vk ) = 0, h, k = 1, 2, 3, . . . . ( 32
)
Equations ( 32) clearly show that the inf uence on the mode m h exerted by the mode e k is measured by the matrix C hk that we can regard as a modal e → m coupling matrix. In an absolutely similar way C kh represents the modal m → e coupling matrix. It is now easy to formulate the following:

CRITERION FOR ELECTRO-MECHANICAL COUPLING. A necessary condition for the presence of electromechanical energy exchange between e k and m h modes is:

C hk = m h , G me (e k ) L 2 = 0.
(33)

Results of numerical simulations

In this section we particularize the evolution equations found in the previous section to the case of a square piezo-electromechanical plate, describing with suitable boundary conditions the action of both the mechanical and electrical constraints. The criterion (33) is exploited in order to establish a self-resonance between the mechanical and electrical vibration modes. The results we f nd in the particular case considered are, however, of wider applicability. Indeed the tuning of the electrical impedence to get a relevant energy exchange between the electrical and mechanical modes is obtained in a form which is independent of both the shape of the plate and the considered boundary conditions. The numerical simulations described in the present section show that the proposed concept of electrically coordinated actuation allow for the eff cient exploitation of available PZT actuators in vibration suppression of relatively large (see table III ) plates.

Analytical solution for the simply supported square plate

Consider a simply supported square piezo-electromechanical plate of side . Assume, moreover, that the electric network interconnecting the PZT actuators -on the boundary -has grounded terminal nodes; in this case the boundary conditions for equations (23) become:

v = 0, Mν = 0, φ= 0, ( 34
)
on each side of the square domain S. The eigenvalues of the purely mechanical and electrical operators:

L mm (f ) = f, L ee (g) = -g, ( 35
) Table I. Mode labeling. k = 1 2 3 4 5 6 7 8 9 i k 1 1 2 2 1 3 2 3 3 j k 1 2 1 2 3 1 3 2 3
are respectively given by:

λ k = π 4 i 2 k + j 2 k 2 , ν k = π 2 i 2 k + j 2 k . ( 36
)
If r denotes the position of the generic material point of S in the reference conf guration, let us introduce the variables x i = (r • e i )/ . Then the eigenfunctions corresponding to the eingenvalues (36) are:

e k = sin(i k πx 1 ) sin(j k πx 2 ), m k = sin(i k πx 1 ) sin(j k πx 2 ), ( 37
)
where the indices i k and j k which determine the modal forms are def ned according to table I.

Recalling that in this case G me (g) = g, we can compute the matrix C hk getting (summation over repeated indices suppressed):

C hk = S (e k )m h = -π 2 i 2 k + j 2 k S e k e h = -π 2 i 2 k + j 2 k δ hk , (38)
where δ hk denotes the Kronecker delta. Note that, because the eigenfunctions are mutually orthogonal, the coupling matrix is diagonal; thus, the coupling can exists only between corresponding modes! The system ( 23) is decomposed into an uncoupled sequence of coupled systems of two equations for two unknown functions like the following:

vh + αλ h v h -γ C hh φh = 0, φh + βν h φ h + δ φh + γ C hh (δv h + vh ) = 0, h = 1, 2, 3, . . . , (39)
or omitting the subscript h:

v + Av -C φ = 0, φ + Bφ + C v + D φ + CDv = 0, (40)
where

A = αλ h , B = βν h , C = γ C hh and D = δ.
Therefore in the case of a simply supported rectangular plate the membrane-like electric network is able to couple one mechanical mode exactly with one electrical mode so that one can get a self-resonance suitably tuning the electric network parameters.

However the parameters appearing in (39) depend on the considered mode number; in general the aforementioned tuning will induce the self-resonance of only one pair of electro-mechanical modes.

Therefore the problem of determining the optimal circuital topology for the interconnection of PZT actuators arises. This problem will be addressed in subsequent investigations. We simply remark here (see also the following figu e 10) that once a self-resonance between a f xed mechanical and a f xed electrical mode is tuned some weaker (but sometimes non negligible and exploitable for damping purposes) electro-mechanical intermodal energy exchanges can be established.

Non damped energy exchange

In order to establish the conditions assuring the maximal energy exchange between the mechanical and electrical states, one could develop the general treatment delineated in the case of one-dimensional electromechanical structures in [START_REF] Vidoli | Modal coupling in one-dimensional electro-mechanical structured continua[END_REF]. However, with a view towards the applications, we consider here a simplif ed version of that treatment, studying the 1-1 coupling through equations ( 40) and then extending the results to multiple couplings.

First of all we analyze the non-dissipative case (D = 0) of equations ( 40). Its solution, starting from purely mechanical initial data v 0 , is the following modulated signal:

v(t) = V 1 cos(α 1 t) + V 2 cos(α 2 t), φ(t) = 1 sin(α 1 t) + 2 sin(α 2 t), ( 41
)
where:

α 1 = 1 2 (C 2 + A + B) -(C 2 + A + B) 2 -4AB , α 2 = 1 2 (C 2 + A + B) + (C 2 + A + B) 2 -4AB , V 1 = u 0 2 1 + C 2 -A + B (C 2 + A + B) 2 -4AB , ( 42
) V 2 = u 0 2 1 - C 2 -A + B (C 2 + A + B) 2 -4AB , 1 = α 2 1 -A α 1 C V 1 , 2 = α 2 2 -A α 2 C V 2 .
In the hypothesis C 2 A, with simple manipulations one can f nd the low frequency analogical components of the modulated signal i.e.:

I MAX = (V 1 + V 2 ) cos α 1 -α 2 2 t , I min = (V 1 -V 2 ) sin α 1 -α 2 2 t , ( 43
)
respectively representing the envelopes of the maxima and minima, and, as usual, related to the energy contents.

Since we are interested to the most eff cient exchange of energy between the mechanical and electrical forms, we seek values of the parameter B that minimize the amplitude of I min and the time T tr elapsed to transform the maximal possible amount of initial energy in electrical form:

min B |I min | = min B |V 1 -V 2 |, min B T tr = max B |α 1 -α 2 |. ( 44
)
These conditions imply respectively:

B 1 = A -C 2 , B 2 = A + C 2 , ( 45
)
indicating that in the interval (B 1 , B 2 ) we get self-resonance. Everywhere in the following we will assume that the self-resonance condition is B = A.

Let us now consider the ratio:

k = C 2 A = g 2 me D P C N , ( 46
)
which is much smaller than 1 in the applications considered here. This number plays an important rôle in determining the pulsation of the low-frequency analog components in the chosen self-resonance condition. Indeed the dimensionless time interval T tr | B=A needed to transform the mechanical energy of the considered mode into electrical energy is given as a function of k (see figu e 4) by:

T tr | B=A = 1 2 1 + √ k -1 - √ k . ( 47
)
Recall that T tr is a dimensionless time and T tr = 1 represents one period of the f rst purely mechanical mode.

The self-resonance condition B = A also implies: Note that β depends on the net-inductance L N that is a tunable parameter. Thus we can simply tune the netinductance value to couple two modes which, verifying the criterion (33), can be made resonant; the optimal value L * Nh to couple the h-th modes is:

β αλ h ν h = απ 2 i 2 k + j 2 k . ( 48
)
L * Nh = 1 (i 2 h + j 2 h )C N 2 ω 2 = 1 (i 2 h + j 2 h )(k ee + k C )N A ω 2 ; (49)
here N A = 2 /d 2 is the total number of actuators.

The energy f ux related to the solution (41) for B = A, is visualized in figu e 5 where the thick gray line represents the electric energy.

Note that in figu e 5 there are four different kinds of energies involved: 1. the mechanical elastic energy, Av 2 /2; 2. the mechanical kinetic energy, v2 /2; 3. the electric inductive energy, Bφ 2 /2; 4. the electric capacitive energy, φ2 /2, only the total sum of these energies is constant.

Damped energy exchange

The characteristic polynomial of equations ( 40) is now:

P (s) := s 2 C 2 + sDC 2 + s 2 + A s 2 + sD + B = 0, (50)
its complex roots representing the damping ratios (real parts) and the pulsations (imaginary parts) of the associated eigenfunctions.

In figu e 6 a pair of roots (the other pair is the complex conjugate) are drawn as functions of the ratio D/C; the gray scale measures the electro-mechanical coupling of the associated eigenvectors: black implies comparable electro-mechanical energy contents. Moreover, the projections of the curves on the planes {D/C, -Re} and {D/C, Im} are drawn.

We observe that: 1. Increasing the ratio D/C (i.e. the net-resistance) def nitively leads to the uncoupling of the electromechanical wave-forms.

2. The projections on the plane {D/C, -Re} show a maximum of the damping ratio of the root relative to wave-form turning to purely mechanical content when D/C tends to inf nity. This circumstance allows for the determination of the critical value for the parameter D proportional to the net-resistance. 3. The projections on the plane {D/C, Im} show that in varying D the eigenfrequencies attain a minimal distance; when D/C tends to inf nity the mechanical eigenfrequency tends to √ A, the electric one vanishes. A further description of the locus of the roots of the characteristic polynomial (50) can be obtained by introducing the following other polynomial:

Q(s) := s 2 C 2 + s 2 + A s 2 + sD + A . ( 51
)
As P and Q are real polynomials their roots can be paired by conjugation having coincident (negative because D is positive) real parts. Let us call these real parts ρ P m , ρ P e and ρ Qm , ρ Qe . It can be proven that:

A > 0, B > 0, D > 0, C > 0 ⇒ (ρ Qm , ρ Qe ) ⊂ (ρ P m , ρ P e ). ( 52
)
Therefore an upper bound for the maximum mechanical damping ratio is obtained by the real parts of the roots of Q when D = 2C and its dimensionless value is 1/2. These considerations are summarized in figu e 7 where the bold lines represent the real parts of the root of Q and the dashed ones represent the real parts of the roots of P when B → A.

Using the approximated condition D = 2C obtained by means of Q and recalling the def nitions of D, C, C hh , and L * N we get the following estimated values for the optimal net-resistance:

R * N = 2π 2 g me C N 3 ω 2 1 M P C N = 2π 2 g me (k ee + k C )N 3/2 A ω 2 1 M P (k ee + k C ) . ( 53
)
Note that R * N is, in this case, independent of the mode number; this fact will not hold true when the clamped plate is considered. Remark that both the values of the optimal inductance L * Nh and resistance R * N decrease when a 'more distributed' net-control system is considered (namely when the number of actuators N A increases). When a dissipative net is considered -in equation ( 40) D = 0 -, the energy, once transformed in electric form, is now dissipated and only a fraction transforms back in mechanical form, this is shown in figu e 8.

In comparing figu e 5 and 8 one should consider that the displayed energy dissipation is obtained for a value of the net-resistance which is not the optimal one.

Numerical solution for the clamped plate

When considering a completely clamped plate a technical diff culty arises in applying the criterion for electro-mechanical coupling. Indeed there is not a treatable closed form for the eigenfunctions of L mm , however one can use their close approximation represented by the product of the eigenfunctions of a clamped-clamped beam (see the f rst row in figu e 9); on the other side we still consider a membrane-like network electrically grounded on its boundary so that the eigenfunctions of L ee will not change (second row in figu e 9). The labeling of the basis in L 2 (S) we have just chosen trivially parallels that previously introduced in table I.

We will apply the coupling criterion to this set of approximated eigenfunctions. The coupling matrix C is no longer diagonal, however, due to the similarities between the modes, the matrix C is quasi-diagonal. Figure 10 shows a representation of the matrix C by means of the gray scale (blank cell means a vanishing value); the f rst nine mechanical and the f rst nine electrical modes are considered.

The novelty in the case of clamped plate is represented by the possibility of coupling one mechanical mode with different electrical ones (if the corresponding C hk is non vanishing). Indeed note that a simple tuning of the net-inductance allows one to make coincident any two frequencies; however all the electric frequencies are shifted together (see figu e 11).

In order to use the analysis of the previous section in the present case, we list in table II:

• the eigenvalues ν k of the spatial operator L ee (g) = g with g = 0 on ∂S;

• the eigenvalues λ k of the spatial operator L mm (v) = v simply supported on ∂S; • the eigenvalues λ (c) k of the spatial operator L mm (v) = v clamped on ∂S; • the ratios c k = λ (c) k /λ k . The eigenvalues λ (c) k have been obtained estimating the Rayleigh ratio on the approximated eigenfunctions. Table II is used to determine the optimal inductance and resistance values, namely:

L * Nh = M P c h (i 2 h + j 2 h )C N π 2 D P , R * Nh = 2g me c h C N D P M P C N . ( 54
)
Figure 12. Plots of the mechanical (black), electrical (gray) and electro-mechanical (dashed) frequency-response functions. The impedance is set to the optimal value for mode 1.

Figure 13. Plots of the mechanical (black), electrical (gray) and electro-mechanical (dashed) frequency-response functions. The impedance is set to the optimal value for modes 2 and 3.

Table III. Mechanical and electrical parameters.

= 1 m h = 1 mm N A = 7 × 7 = 49 g me = 28.10 -5 N V -1 ρ = 2700 Kg m -3 E Y = 70 GPa k ee = 0.6 µF k C = 1 µF
We conclude this section with two plots of the frequency-response functions. In them the net-inductances and resistances are tuned respectively on the optimal values (L * N1 , R * N1 ) and (L * N2 , R * N2 ). The black and gray lines represent the norms of the purely mechanical and purely electric part of the response function, while the dashed one represents the norm of the coupling part of the response function.

The numerical results, shown in figu es 12 and 13, conf rm the validity of the coupling criterion as a consequence of the optimal impedance values. Indeed the electro-mechanical coupling shows maximal peaks with wider frequency bandwidths when these optimal values are chosen.

In the numerical simulations the values in table III were used. The assumed performances of PZT actuators are realistic; they concern the ACX actuators QP20W.

Response of piezo-electromechanical plates to impulsive external actions

In this subsection we describe some numerical simulations conceived in order to describe the response of piezo-electromechanical plates to a large class of impulsive external actions. These simulations allowed us to produce some interesting and suggestive animations. The interested reader can download them from the web-site http://www.esm.vt.edu/∼henneke/vcpa/animation.

We considered a clamped piezo-electromechanical plate, electrically grounded at its boundary. This last boundary condition means that the electric network interconnecting the PZT actuators behaves as an edgesupported membrane.

Moreover it is assumed that only the modes previously labelled (1, 1), (1, 2), (2, 1), (2, 2) (see table I ) can be considered in Fourier expansion and that initial displacement and electric state are everywhere vanishing. Finally the network is tuned choosing as optimal impedence that found for mode (1, 1).

The impulsive external action is represented by a Dirac δ initial velocity f eld concentrated in the two points P 1 and P 2 of the piezo-electromechanical plate. In the case of vibrations described in figu es 14 and 15 the point P 1 has coordinates (0.6L, 0.55L), while those in figu es 16 and 17 are relative to external actions applied in P 2 = (0.75L, 0.5L) where L is the length of the plate side.

The numerical experiment shows that in the f rst case nearly all electromechanical vibration energy is used to excite the f rst mode. The black dot represents the localization of P 1 .

The corresponding time evolution of both mechanical displacement and electric state is shown in figu e 15, which clearly shows an eff cient transfer of energy from mechanical into electrical form.

Rather different is the result represented in the subsequent f gures: the impulsive action being concentrated in P 2 , a non-negligible amount of electromechanical vibration energy is used to excite also the second mode. As the optimal impedences are tuned on the f rst mode, and as the coupling coeff cient between the f rst and second mode is very small (see figu e 10), the mechanical vibrations cannot be dissipated using the piezoelectric action of the actuators. This was expected as the electromechanical coupling matrix is nearly diagonal and therefore only a f xed pair of electrical and mechanical modes are coupled. The previous considerations are conf rmed by the time evolutions plotted in figu e 17, the mechanical vibration energy is only reduced by the PZT actuation, while after a characteristic time the electric vibration fades. This means that only the mechanical energy initially exciting the f rst mode is transferred into an electric vibration mode and then dissipated. The non-negligible part of mechanical energy present in the second mechanical mode is not transferred into any electrical mode and is not therefore dissipated.

We conclude that in order to further improve the eff ciency of the considered device, more complicated interconnecting electric networks need to be conceived.

Conclusions

The device proposed in this paper is based on the concept of global synergic response of a set of PZT actuators to a given mechanical vibration mode. This global response is obtained by conceiving an electric network interconnecting the single actuators. The single electric signal produced by one of them is the potential drop between a node of the electric net and the ground. Thus, PZT actuation is accompanied by an electric waveform which evolves together with the mechanical one. In order to study the performances of the conceived system a mathematical model of its dynamic behavior has been developed. The model is obtained by means of a homogenization procedure and therefore it gives only rough predictions when short wavelengths of the electromechanical signals are considered. In the framework of this model, it is proven that a criterion exists assuring electro-mechanical coupling. This criterion allows for the determination of the net-impedances maximizing the electromechanical energy exchange. The eff ciency of the device is indicated by the very low damping ratios which it shows when the optimal net impedance is chosen. This is its main advantage when compared with the devices based on the concept of concentrated actuation (see [START_REF] Guran | Wave Motion, Intelligent Structures and Nonlinear Mechanics[END_REF] and [START_REF] Fuller | Active Control of Vibration[END_REF]). Furthermore a remarkable decrease of the needed impedance for electro-mechanical coupling is obtained.

At the present stage the construction of a prototype to prove that the concept proposed can be realized in practice, -standing the available technological state of the art -is needed. Two experimental set-ups are being developed at Virginia Polytechnic Institute and State University and Università di Roma La Sapienza.

Figure 1 .

 1 Figure 1. Assembled plate and network.

Figure 2 .

 2 Figure 2. Electrical connection scheme.

Figure 3 .

 3 Figure 3. Low-frequency analogue components.

Figure 4 .

 4 Figure 4. Elapsed time to transfer energy T tr as a function of the coupling coeff cient k.

Figure 5 .

 5 Figure 5. Electric (gray) and mechanical (black) energies vs. time.

Figure 6 .

 6 Figure 6. Characteristics roots as functions of the ratio D/C.

Figure 7 .

 7 Figure 7. Plot of the polynomial Q(s).

Figure 8 .

 8 Figure 8. Dissipation of electric (gray) and mechanical (black) energies.

Figure 9 .

 9 Figure 9. Functional basis for the clamped plate.

Figure 10 .

 10 Figure 10. Representation of the coupling matrix. A blank cell indicates a vanishing value.

Figure 11 .

 11 Figure 11. Electric and mechanical frequencies.

Figure 14 .

 14 Figure 14. Mechanical (left column) and electrical (right column) vibrations induced in the piezo-electromechanical plate by an impulse concentrated at the point P 1 . Its localization is specifie by the black dot. Electric potential is represented as a black-white scale: gray means ground, non-dimensional time is indicated.

Figure 15 .

 15 Figure 15. Time evolution of displacement and electric state in a representative point of the piezo-electromechanical plate, with impulse concentrated in P 1 .

Figure 16 .

 16 Figure 16. Mechanical (left column) and electrical (right column) vibrations induced in the piezo-electromechanical plate by an impulse concentrated at the point P 2 . Its localization is specifie by the black dot. Electric potential is represented as a black-white scale: gray means ground, non-dimensional time is indicated.

Figure 17 .

 17 Figure 17. Time evolution of displacement and electric state at a representitive point of the piezoelectromechanical plate, with impulse concentrated at P 2 .

Table II .

 II Compared eigenvalues.

		k = 1	2	3	4	5	6	7	8	9
	ν k = (i 2 k + j 2 k ) λ k = (i 2 k + j 2 k ) 2 λ (c) k	2 4 13.4	5 25 55.8	5 25 55.8	8 64 121.6	10 100 180.2	10 100 180.2	13 169 282.6	13 169 282.6	18 324 501.0
	c k	3.35	2.23	2.23	1.90	1.80	1.80	1.67	1.67	1.55

* Correspondence and reprints: francesco.dellisola@uniroma1.it

More complicated and maybe more efficien circuital topologies are conceivable: this being the object of further investigations.

In the following integrals the standard notation used in modern measure theory is used. Therefore for instance the infinite imal volume dV is not indicated. As the integration domains are indicated no misunderstanding is possible.

This choice implies that the fir t purely mechanical mode in the case of a simply supported plate has a pulsation equal to 2π .

Acknowledgments

This research was partly performed when the authors were visiting the Department of Engineering Sciences and Mechanics, Virginia Polytechnic Institute and State University, the warm hospitality was greatly

appreciated. They gratefully acknowledge Professor E. Henneke, Head of ESM department, and Professor R. Batra for having secured visiting grants. We need to thank Mark D. Carrara M.S. for his help in revising the English form of the paper.