F Ben Slimen

Nathalie Gaumer

Stéphane Chaussedent

M Haouari

H Ben Ouada

E-Mrs Spring Meetin

Optical and structural properties of Eu3+ doped silicophosphate glasses

come   L'archive ouverte pluridisciplinaire

) Wavelength(nm)

S2

The fluorescence line-narrowing spectra of 1mol% Eu 3+ doped sample of sol-gel silicophosphate with Si/P/Eu=1/0.1/0.01 molar ratio densified at 900°C was investigated. The spectra were obtained by selective excitation within the inhomogeneously broadened 7 F 0 → 5 D 0 absorption band. Intensities are normalized to the most intense 5 D 0 → 7 F 1 peak.

The lifetimes of 1mol% Eu 3+ doped sample was measured at green excitation ( 532 nm) and at different excitation wavelengths along the 7 F 0 → 5 D 0 absorption band. The luminescence was collected at 612 nm within the 5 D 0 → 7 F 2 band and measurements were performed at 77K. The results show a two-step behavior as excitation wavelength increases which supports the existence of at least two broad varieties of Eu 3+ sites.

The average peak position energies of the three 7 F 1 bands of Eu 3+ in silicophosphate glass are plotted as a function of the excitation wavelength within the 7 F 0 → 5 D 0 excitation band.

The crystal field strength parameter decreases with the increase of the excitation wavelength but with a change in the slope. That is consistent with the results obtained by the decay measurements.

S 2 = 2 20 2 2 22 22 (E 3 -E 1 ) 20 5/3 E 3 E 1 2E 2
22 , 20 : Crystal field's parameters. E 1 , E 2 , E 3 : Energies of the three 7 F 1 bands of Eu 3+ .

The strong decrease of the average crystal parameter S 2 suggests a large variation in the local environment for the Eu 3+ ions in this glass. It's shows an intermediate behavior between a pure phosphate and a pure silicate. This combination of properties can be understood by taking into account the mixed nearestneighbor coordination, as the influence of the local environment in the Eu 3+ spectra not only depends on the geometrical changes of coordination but also on the nature of the coordinating species. [3] The effect of phosphate codoping in europium-doped silica glasses was investigated using both FLN technique and decay times at different excitation wavelengths. The wavelength dependence of lifetimes shows a two-step behavior as excitation wavelength increases which supports the existence of at least two broad varieties of Eu 3+ sites. This result is also confirmed by the crystal parameter S 2 analysis . . The study of other concentrations is in progress.

[1]: A. Monteil, S. Chaussedent, G. Alombert-Goget, N. Gaumer, J. Obriot, S.J.L. Ribeiro, Y. Messaddeq, A. Chiasera, M. Ferrari. Journal of Non-Crystalline Solids 348. 44-50 (2004) .

[2]M.Kamal, I.K.Battisha, M.A.Salem., Journal of Sol-Gel Sci Technol, 58. 507-517 ( 2011).

[3]: R.Balda , J. Fernandez, J. L. Adam, M.A. Arriandiaga. Physical review B, 54.17 ( 2006).


Denis Pitzalis

Geneviève Aitken

Florent Autrusseau
email: florent.autrusseau@univ-nantes.fr

Marie Babel
email: mbabel@insa-rennes.fr

François Cayre
email: cayre@gipsa-lab.inpg.fr

William Puech
email: william.puech@lirmm.fr

D Pitzalis¹

G Aitken¹

F Autrusseau²

M Babel³

TSAR: Secure Transfer OF High Resolution Art Images

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Cultural Heritage (CH) institutions, in particular museums, are supposed to undertake at least two essential missions [START_REF] Xiaowu | Ming University Digital Museum Grid[END_REF]. Firstly, they have to preserve the items that represent our past and our production and save them from any damage in order to make them available for the future. At the same time, these institutes play an active role in the spread of cultural knowledge. This last educational objective imposes them to widely show and communicate these materials. However, these two missions are somewhat contrary in nature because handling works of art inevitably causes damage. To help solving this major problem, lot of investigations and researches go in the direction of "Digital Libraries" [2,3,4]: digital versions of the original art items are collected in a database on a server accessible via the Internet.

For example, the National Gallery in London, the Tokyo University Digital Museum (through the Digital Museum 2000 project) and the Louvre Museum provide public access to their databases. However, users can only browse, visualize and download low-resolution images. The application has been actually designed to prevent illegal copies of digitized data and the best and easiest way to achieve this consists basically in not transmitting the high resolution content (scientific photography, document scans, etc). High resolution information is therefore stored separately and available for a limited number of people. This is also the case, in France, of the C2RMF laboratory, based in the Louvre museum. During the last years, it has collected digital information of more than 60,000 works of art concerning more than 1200 French museums. Part of this digital information consists of high EVA 2008 Florence resolution images (up to 20,000 x 20,000 pixels). This information, gathered in the EROS database, is for the moment only accessible to researchers whose work is connected with the C2RMF's activities. In order to open the database to a wider public, when still assuring the protection of the sensible data, two problems have to be resolved. First, as the amount of data is huge, an efficient compression method has to be applied. In particular, for research purposes, this compression scheme has to provide a lossless solution. Secondly, the application must support interactive actions that can make browsing easier for users, together with efficient protection policy.

This article is organized as follows. In the next section the TSAR project and the EROS database are described as the context of our study. After the compression methods are presented. Then we describe the visible and invisible watermarking solution, the encrypting techniques. In the conclusion we show the principles of the corresponding client-server application integration.

THE TSAR PROJECT

The TSAR project (Secure Transmission of high-Resolution Art images) is supported by the French National Research Agency. Five laboratories are involved, namely the C2RMF (Louvre, Paris), IETR (Rennes), IRCCyN (Nantes), LIRMM (Montpellier) and LIS (Grenoble). The C2RMF has developed the EROS (European Research Open System) database storing digital art documentation [START_REF] Aitken | Database Management and Innovative Applications for Imaging within Museum Laboratories[END_REF].

The idea is to create a framework to integrate the different security solutions in order to secure the access to the images. The objective is to build hierarchical secure protection based on the rights of various user groups (figure 1). frared, multispectral, 3D etc). It even has a particle accelerator for ???????? pu y of work and few tera-bytes of digital content, mainly composed of high resolution images.

HE EROS DATABASE T

The C2RMF is today one of the most active labs in the field of research in cultural heritage at an international level. The laboratory performs not only restoration, but also documentation and scientific studies, especially chemical and physical analysis, advanced scientific imaging (X-ray, in rpose.

The C2RMF has been an active participant in many EU projects since 1989. In this context, many projects have tried to build on and extend the work carried out in previous works, resulting in a coherent and inter-dependent bod EVA 2008 Florence These images are accessible by an image server internally developed that is able to provide a quick access to huge images [START_REF] Pitzalis | A new Concept in high Resolution Intern Browsing[END_REF]. This system, called IIPImage, is designed for the remote viewing of very high resolution images across the Internet. Its architecture gives it the peculiarity to be usable even over a slow dial up connection. The server can be launched as a plug-in that can work with a variety of web servers such as Apache, Lighttpd, MyServer, Microsoft IIS or any other FCGI-enabled http servers.

Images can be browsed via feature-rich Java or Flex client, a simple Javascript client embedded within a web page or downloaded as a JPEG image dynamically generated at the requested size and resolution by the server. The modular structure of the server gives the possibility to integrate all the security technologies developed within the TSAR consortium to protect the image transmission at different levels.

DATA COMPRESSION METHODS

Advances in digital imaging have resulted in massively increased image resolutions and bit depths as well as in new techniques such as 3D imaging and topographical scanning. These new imaging techniques produce huge quantities of data that need to be adequately managed. Because of the sheer volume of data, new image optimized lossless compression methods must also be investigated.

LAR compression

The LAR method has been initially introduced for lossy image coding [START_REF] Ronsin | Color lar codec: a color image lfdings nd e, France, 2006. r ics, Imaging and Vision[END_REF]. It is based on the assumption that an image can be represented as layers of basic information and local texture. Thus, the overall scheme of this approach consists of two scalable layers: a first one to encode an image at low bit rates, and a second one for visual quality enhancement at medium/high bit rates. The method relies first on Quadtree partitioning of the image. Homogeneity is provided by a local gradient estimation that fixes small blocks onto contours and large ones onto flat areas. The Quadtree Partition is denoted as QP [Nmax...Nmin] with all square blocks having a size equal to a power of two, and N max and N min representing respectively the maximum and minimum authorized block sizes. The first layer representation consists of rebuilding each block by its mean luminance value, encoded using a DPCM approach. The low bit-rate rebuilt image is visually acceptable thanks to the Quadtree Partition, that accommodates the variable blocksize as a function of the original image context. To encode texture, the second layer performs a variable blocksize DCT transform to fit the QP [Nmax...Nmin] partition. Scalability can then be content-dependent: the enhancement of contours presupposes that only the smallest blocks are processed. Subjective quality tests have been carried out between JPEG, JPEG2000 and LAR codecs: LAR outperforms other techniques for nearly all the test sequences. For color images, an original hierarchical region-based representation technique adapted to the LAR coding method has been designed, allowing advanced functionalities such as Region of Interest coding or region-level encoding of chromatic images (Figure 2). The LAR compression method can also be used, by its nature, as crypting algorithm [START_REF] Babel | Context-Based Scalable Coding and Representation of High Resolution Art Pictures for Remote Data Access[END_REF].

Mojette compression

The baseline of this compression method consists in encoding the difference between several similar finite Radon projections. Two differential encoding techniques are required to efficiently encode the Radon transformed data. An intra-projection encoding technique is first performed. This differential encoding process takes benefit of the redundancy present in such projections. The second step consists of encoding the similarities between Radon projections, this is the inter-projection encoding procedure.

The used Finite Radon transform (called Mojette transform [START_REF] Autrusseau | Lossless Compression Based on a Discrete a Exact Radon Transform: A Preliminary Study[END_REF]), allows to ensure both compression and data redundancy. The main novelty of this work is the use of a secure distributed storage or transmission tool in a lossless compression context (figure 3). is i-171 and ii-172. O yed as images with columns heights of q i and a width of 514 ATERMARKING elective encryption Figure 3. In the center two 1D projections with projection vectors (1, q i ) where q i n the right the same projection data displa

W S

The proposed approach is a selective encryption in the Huffman coding of the Discrete Cosine Transform (DCT) coefficients using the Advanced Encryption Standard (AES) [START_REF] Rodrigues | A selective encryption for heterogenous colo JPEG images based on VLC and AES stream cipher[END_REF].

EVA 2008 Florence

Selective encryption (SE) is a method that encodes only the most important portion of the data in order to provide a proportional privacy and to reduce computational requirements. The objective of our work is to leave free the low-resolution image and give full-resolution access only for authorized person. This approach is based on AES stream ciphering using VLC (Variable Length Coding) of the Huffman's vector. The proposed scheme allows decryption of a specific region of the image and results in a significant reduction in encrypting and decrypting processing time. It also provides a constant bit rate and keeps the JPEG bit-stream ompliance (figure 4).

UALITY EVALUATION

lity assessment should include special requirements that depend on the ent, and a spatial pooling taking into account the density and the structure of the error.

ONCLUSION AND FUTURE WORKS

S

More investigation is done in the watermarking field with two other methods that must be truly secure in the WOA framework [START_REF] Cayre | Kerckhoffs-based embedding security classes for woa data hiding[END_REF]. The first one is called Natural Watermarking and can be made either stego-secu w

Q

In watermarking and data hiding context, it may be very useful to have methods checking the invisibility of the inserted data or, at least, checking the objective quality after the mark embedding or after an attack on the watermarked media [START_REF] Callet | Visibility control and Quality assessment of watermarking and data hiding algorithms[END_REF]. Many works exist in the literature dealing with quality assessment mainly focused on compression application. Nevertheless, visual qua application context. Komparator is an objective quality metric for color images based on Human Visual System (HVS) properties which doesn't need any priori knowledge about the type of degradation introduced in the distorted image. As first step, a visual representation of the original and the distorted images is computed: in this stage a model based on results obtained by psychophysics experiments on color perception and on masking effects is used. The model adopted for early vision stages is inspired by the S. Daly's Visual Difference Predictor (VDP) extended to color. The visual representation stage is then followed by an error pooling process merging all the obtained distortion maps into a single value. This stage is made up by a frequency, a compon

C

EVA 2008 Florence

The opening of art picture databases represents a promising method for the spread of cultural knowledge but it demands a number of requirements in terms of reproduction rights and access control, along with a fast, secure and user-friendly navigation. es.

Cultural Heritage institution with a real set of pplication is currently underway and will be the result of the arallel works ongoing in the TSAR project. This paper has presented an overview of a first conjoint effort, in the TSAR project, based on multiple parallel research projects in terms of efficient and original solutions for secure transmission of high definition imag

The LAR, the Mojette transform, the selective encryption, the persistent watermarking methods are developed all together within a case studies and a real problematic.

The design of a client-server a p

Figure 1 .

 1 Figure 1. The TSAR/C2RMF adopted solution. A dynamic managed access to image repository.

Figure 2 .

 2 Figure 2. Low resolution LAR image and hierarchical region representation on "CRT" with partition QP[16...2] and N0 < 2

Figure 4 .

 4 Figure 4. a) Work-of-mage for C = 128, c) Ciphered image for

  c art original image 512×640, b) Ciphered i C = 8 ????.

  pported by the French National Research Agency as part of the TSAR project (ANR ARA SSIA TSAR 2005 -2008). Re ji, Z. Yi, H. Ke and M. Shunag, "The Progress of oceedings IEEE International Conference on [2]

ACKNOWLEDGMENTS

This work is su ferences


Editor K De

Meyer C Durand

F Casset

P Renaux

D Renaud

E Ollier

B Legrand

L Buchaillot

A M Ionescu

I. INTRODUCTION

F OR THE first time, an in-plane nanoelectromechanical (NEM) resonator based on a resonant suspended gate (RSG) MOSFET principle [START_REF] Abelé | Suspended-gate MOSFET: Bringing new MEMS functionality into solidstate MOS transistor[END_REF], [START_REF] Grogg | Integration of MOSFET detection in MEMS resonators for improved output detection[END_REF] and integrated in a front-end process is demonstrated. Advanced silicon-on-nothing (SON) technology [START_REF] Monfray | First 80 nm SON (Silicon-On-Nothing) MOSFETs with perfect morphology and high electrical performance[END_REF] based on industrial 8-in tools is used to fabricate RSG-MOSFETs with high in-IC integration capabilities.

This letter reports on the design, fabrication process, measurement setup, and experimental results of a 14-MHz NEM resonator and compares a classic capacitive detection [START_REF] Durand | Characterization of In-IC integrable in-plane nanometer scale resonators fabricated by a silicon on nothing advanced CMOS technology[END_REF] with a MOSFET-based detection. Operational dc and vibrating characteristics of laterally excited nanogap RSG-MOSFETs are also reported for the first time. The proposed process and fabricated resonators enable the in-IC integration of off-chip modules such as time reference functions with significant cost, size, and consumption reduction [START_REF] Nguyen | Vibrating RF MEMS technology fuel for an integrated micromechanical circuit revolution?[END_REF].

II. DEVICE DESIGN AND FABRICATION

The lateral MOSFET is composed of a vibrating gate and a channel separated from the mobile gate by an insulator and an air gap (Fig. 1). Drain and source are placed in the same horizontal plane as the gate. The mobile-gate (beam) vibration at resonance frequency modulates the mobile charges of the MOS conductive channel formed along the lateral etched sidewall and, consequently, the drain current. The design and the proposed process enable the integration of both capacitive [START_REF] Durand | Characterization of In-IC integrable in-plane nanometer scale resonators fabricated by a silicon on nothing advanced CMOS technology[END_REF] and amplified detections using the MOSFET intrinsic gain (tailored by the transistor transconductance i ds = ν gs × g m ).

Devices were fabricated by using the SON technology [START_REF] Monfray | First 80 nm SON (Silicon-On-Nothing) MOSFETs with perfect morphology and high electrical performance[END_REF] to achieve sub-100-nm gaps and 400-nm-thick single-crystal silicon resonators using only front-end processes [START_REF] Durand | Characterization of In-IC integrable in-plane nanometer scale resonators fabricated by a silicon on nothing advanced CMOS technology[END_REF], [START_REF] Durand | Silicon on nothing MEMS electromechanical resonator[END_REF] and materials, ensuring the in-IC integration capabilities. The fabrication-process flow presented in [START_REF] Durand | Characterization of In-IC integrable in-plane nanometer scale resonators fabricated by a silicon on nothing advanced CMOS technology[END_REF] and [START_REF] Durand | Silicon on nothing MEMS electromechanical resonator[END_REF] is resumed in Fig. 2. It starts with the patterning of active areas through a thermal SiO 2 layer. A SiGe sacrificial layer is then grown by selective epitaxy and followed by a low boron-doped (10 16 at/cm 3 ) nonselective single-crystal silicon epitaxy. Phosphorous dopants are implanted (10 19 at/cm 3 ) to define gate, 0741-3106/$25.00 © 2008 IEEE source, and drain. E-beam lithography is used to define gaps and the resonator structure, leading to a 47-nm gap resolution. The 400-nm-thick silicon structural layer is then etched by an anisotropic plasma to define air gaps. After dopant diffusion and activation annealing steps, structures are released by an isotropic plasma etch of the SiGe sacrificial layer. The released structures were protected by a nonconformal SiO 2 deposition to allow metal pad deposition. Fig. 2 shows a picture of a suspended RSG-MOSFET.

III. MEASUREMENT SETUP

RF characterizations are performed by using an Agilent 8753E network analyzer. Fig. 1 shows the two measurement configurations for comparing capacitive and MOSFET detections on the same device.

For a capacitive-detection measurement [START_REF] Durand | Characterization of In-IC integrable in-plane nanometer scale resonators fabricated by a silicon on nothing advanced CMOS technology[END_REF], the transmitted signal through the resonator is measured between the vibrating beam and the electrode of the device. A bias voltage V dc is applied on the electrode; the beam and the substrate are grounded to avoid any pull-in effect of the structure.

For a MOSFET-detection measurement, three bias voltages are applied: the electrode V dc , the gate voltage V g , and the drain voltage V ds . Optimal V g and V ds values for dynamic characterization were extracted from the static MOSFET characteristics I d (V g ) shown in Fig. 3 as well as I d (V ds ).

IV. EXPERIMENTAL RESULTS

Fig. 3 shows the static electrical I d (V g ) measured characteristics of the MOSFET transistor. The threshold voltage V th is 2 V. An OFF-state leakage current of 0.1-1.3 µA depending quasi-linearly on V ds is observable at V g = 0 V. It is attributed to a photolithography misalignment when protecting channel from phosphorous implants. Indeed, channel surface was partially implanted on a few nanometer widths, generating a 2.1-MΩ short-circuit resistor in parallel of the transistor, explaining the high leakage current and its quasi-linear dependence on V ds . Optimal operating points were defined from static characteristics: V g being 4.6 V corresponding to the maximum of transconductance and V ds being 3 V corresponding to the saturation region, with an OFF-state leakage of 1.4 µA.

Fig. 4 compares the resonant amplitude response between capacitive and MOSFET detections, on the same structure (L = 10 µm, w = 165 nm, and d = 120 nm). The fundamental resonance frequency was measured to be 14.33 and 14.31 MHz with capacitive and MOS detections, respectively. The slight discrepancy between the two measurements is explained by the different bias conditions of actuation between the MOS and capacitive detections. The frequency shift is mainly due to the MOSFET channel surface potential which is set to zero in the case of the capacitive detection. Measurements are in good agreement with mechanical analytical calculations and FEM simulations, giving 14.43 and 14.42 MHz, respectively, being independent of the detection principle. The extracted motional resistance R m is 736 kΩ, and the C 0 capacitance is 21.5 fF. The MOSFET detection yields a +4.3 dB signal amplification compared with the capacitive detection due to the MOSFET intrinsic gain. This amplitude could be increased first by optimizing the transistor design (channel width/length ratio), hence enabling a higher g m , and second by limiting the leakages. In this way, the SiGe layer has to be completely etched, and the SiO 2 insulation layer has to be thicker. Fig. 5 shows the influence of the beam polarization on the output signal using the MOSFET detection. Its amplitude increases with V dc due to the larger beam vibration amplitude caused by the larger electrostatic excitation. The effect is also reinforced by the air-gap reduction induced by the increase in V dc . On the other hand, the electrostatic field increases with V dc and acts as a negative stiffness, shifting down the resonant frequency of 10% for a 5.5 V V dc variation. The influence of V ds on the device response was also investigated. Results show an increase in the resonant peak amplitude with V ds , as predicted on static characteristics for a given V g of 4.6 V, due to the increase in g m with V ds . Moreover, a V ds variation from 2.0 to 3.5 V induces a 0.16% decrease in the resonant frequency. This effect is negligible regarding the shift induced by V dc .

Dependence of resonators' behavior with environmental conditions was studied in [START_REF] Durand | Characterization of In-IC integrable in-plane nanometer scale resonators fabricated by a silicon on nothing advanced CMOS technology[END_REF] on similar devices. Resonant frequencies shift down quasi-linearly with temperature (from -50 • C to 100 • C), giving a temperature coefficient of frequency of -32 ppm/ • C. Quality factors were found to be maximum and constant at low pressure (< 1 Torr), whereas they decrease at higher pressures, showing that even for nanoscaled beams, air damping remains the major source of dissipation.

V. CONCLUSION

We fabricated and experimentally demonstrated, for the first time, a fully operational in-plane 14-MHz RSG-MOSFET resonator with intrinsic tunable amplification. The unique large gain, in terms of peak detection amplitude, makes this new device ideal for any type of NEM structure displacement detection. Future works will focus on reliability issues such as fatigue [START_REF] George | Low-cycle fatigue in silicon: Comparison with fcc metals[END_REF], packaging, and in-IC integration for industrial perspectives.

  In-Plane Silicon-On-Nothing Nanometer-Scale Resonant Suspended Gate MOSFET for In-IC Integration Perspectives C. Durand, F. Casset, P. Renaux, N. Abelé, B. Legrand, D. Renaud, E. Ollier, P. Ancey, A. M. Ionescu, and L. Buchaillot Abstract-A 14-MHz in-plane nanoelectromechanical resonator based on a resonant-suspended-gate (RSG) MOSFET principle and integrated in a front-end process is demonstrated. The devices are in-plane flexural vibration mode beams (L = 10 µm, w = 165 nm, and h = 400 nm) with 120-nm gaps. This letter details the design and process flow fabrication steps. Then, the electrical device characteristics are demonstrated, comprising static and dynamic studies around the resonant frequency. Devices enable the comparison of a pure capacitive detection with the RSG-MOSFET-based detection on the same component, showing a 4.3-dB-huge peak. Due to its output signal amplification and in-IC integration potentialities, the RSG-MOSFET-based detection is ideal for any type of nanoelectromechanical structure displacement detection. Index Terms-In-IC integration, in-plane resonator, nanometerscale resonator, resonant suspended gate (RSG) MOSFET, silicon on nothing (SON).

Fig. 1 .

 1 Fig. 1. Principle schematic and electrical characterization setup of the RSG-MOSFET device.

Fig. 2 .

 2 Fig. 2. Perspective SEM picture of a beam (L = 10 µm, w = 165 nm, and d = 120 nm).

Fig. 3 .

 3 Fig. 3. Static MOS characteristics I d (Vg) and Gm(Vg).

Fig. 4 .

 4 Fig. 4. Capacitive-and MOSFET-detection comparison on the dynamic response of the NEMS (L = 10 µm, w = 165 nm, and d = 120 nm).

Fig. 5 .

 5 Fig. 5. Influence of V dc variation on the dynamic response of the NEMS (L = 10 µm, w = 165 nm, and d = 120 nm) using the MOSFET detection.


B Bergeot

S Bellizzi
email: bellizzi@lma.cnrs-mrs.fr

Dynamic instability mitigation by means of nonlinear energy sinks in mechanical systems having one or two unstable modes

Dynamic instability mitigation by means of nonlinear energy sinks in mechanical systems having one or two unstable modes

Baptiste Bergeot, S Bellizzi

To cite this version: In the context of passive mitigation of dynamic instabilities, it is now established that the Nonlinear Energy Sinks (NESs) are good candidate to consider especially when low frequency and high level are concerned [START_REF] Gendelman | Asymptotic analysis of passive nonlinear suppression of aeroelastic instabilities of a rigid wing in subsonic flow[END_REF][START_REF] Luongo | Aeroelastic instability analysis of nes-controlled systems via a mixed multiple scale/harmonic balance method[END_REF]. The operation of the NESs is based on the concept Targeted Energy Transfer (TET). A basic NES generally consists of a light mass, an essentially nonlinear spring and a viscous linear damper. Because of its essentially nonlinear stiffness, a NES can engage in resonance over a broad frequency range. Whether for a system under impulsive, harmonic or broadband frequency excitation or whether for an auto-oscillating system, TET results from nonlinear mode bifurcations. In general, the phenomenon of TET can be described as a 1:1 resonance capture [START_REF] Gendelman | Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian Systems[END_REF].

This work considers auto-oscillating systems as primary system and it is divided into two parts described below.

1 Prediction of steady-state regimes of a multi-degree-of-freedom unstable dynamical system coupled to a set of nonlinear energy sinks Here the primary system is a multi-degree-of-freedom (multi-DOF) unstable mechanical systems undergoing cubic nonlinearities and coupled to M NESs. We assume in this section that the primary system has only one mode which can become unstable through Hopf bifurcation.

We propose an analytical method to predict the steady-state regimes of the coupled system. The method begins using the biorthogonal transformation to diagonalize the primary system written in the state-space form. Afterwards, the dynamics of the diagonalized system is reduced keeping only the unstable mode and ignoring the stable modes. Then the slow-flow of the system is obtained using the complexification-averaging (CA-X) method within the assumption of a 1:1 resonance capture around the frequency of the unstable mode. The resulting slow-flow possesses a small parameter related to the mass of the NES and, in the framework of Geometric Singular Perturbation Theory (GSPT) [START_REF] Jones | Geometric singular perturbation theory[END_REF], it defines a (2M, 1)-fast-slow system. The slow variable characterizes the unstable mode of the primary system whereas the 2M fast variables describe the NESs motions (amplitude and phase). It is shown that the Critical Manifold (CM) of the slow-flow can be reduced to a one dimensional parametric curve evolving in a multidimensional space. A similar form of the CM is obtained considering a network of parallel NESs [START_REF] Bergeot | Asymptotic analysis of passive mitigation of dynamic instability using a nonlinear energy sink network[END_REF]. The knowledge of the stability properties of the critical manifold and the fixed point of the slow-flow (position and stability) makes it possible to predict the response regimes. Finally, the method is applied to the prediction of the steady-state responses of an airfoil undergoing an aeroelastic instability coupled the a set of NESs (from one to four). Theoretical results are compared, for validation purposes, to direct numerical integration of the system. The comparison shows a good agreement.

Mitigation by means of nonlinear energy sinks of friction-induced vibrations in a mechanical system having two unstable modes

In this part, the primary system has two unstable modes and, as a first step, it is linear and coupled to only one NES. Our objective is to investigate the solutions in the vicinity of two simultaneous 1:1-1:1 resonances to the natural frequencies of the two unstable modes. The procedure described above is performed again except that now the two unstable modes must be kept. Moreover, in spite of the presence of the NES coupling, we assume that each variable associated to the primary system (resulting to the biorthogonal transformation) oscillates at one single frequency contrary to the degrees of freedom associated to the NES which must be split as a sum of two terms to capture frequency components with respect to the two unstable modes. Within this assumptions, the CA-X method is applied leading to a slow-flow (in the real domain) which takes the form of a (4, 2)-fast-slow system. The slow variables characterize the two unstable modes of the primary system whereas the 4 fast variables describe the NESs motion (amplitude and phase of the two frequency components).

Preliminary analytical results, again based on GSPT, are presented. For example the CM, which appears as a 6-dimensional parametric surface, is determined as well as fixed points of the slow-flow and their stability. These results are validated by comparison to numerical simulation of the system.

  Bergeot, S Bellizzi. Dynamic instability mitigation by means of nonlinear energy sinks in mechanical systems having one or two unstable modes. Seventh International Conference on Nonlinear vibrations, Localization and Energy transfer, Jul 2019, Marseille, France. hal-03208406

th International Conference on Nonlinear Vibrations, Localization and Energy Transfer July 1-4, 2019 -Marseille, France


Baptiste Bergeot

Sergio Bellizzi

come

Control of helicopter ground resonance by means of passive non-linear energy sink

LMA, CNRS UPR7051, Aix-Marseille Univ., Centrale Marseille, F-13402 Marseille Cedex 20, France a baptiste.bergeot@centrale-marseille.fr b bellizzi@lma.cnrs-mrs.fr c bruno.cochelin@centrale-marseille.fr

Extended Abstract

Problem Statement. Helicopter Ground Resonance (HGR) [START_REF] Coleman | Theory of self excited mechanical oscillations of helicopter rotors with theory of self excited mechanical oscillations of helicopter rotors with hinged blades[END_REF][START_REF] Malburet | Instabilité mécanique: contrôle actif et passif, chapter 2[END_REF] is a dynamic instability involving the coupling of the blades motion in the rotational plane (i.e. the lag motion) and the motion of the fuselage. HGR can be very violent and can lead to the total destruction of the aircraft. The present work is a preliminary study of the capacity of a passive nonlinear absorber (or Nonlinear Energy Sink (NES)) to control an HGR.

Framework. In the domain of passive control vibration, it has been shown that the use of a NES can provide very interesting results. Under certain conditions an irreversible energy transfert from the primary system to the NES can occur. This phenomenon is called Targeted Energy Transfert (TET). In their seminal paper [START_REF] Gendelman | Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian Systems[END_REF], Gendelman et al. analyze TET in term of resonance capture and nonlinear modes. More recent works show that the use of NES is also an interesting way to control dynamic instabilities. For instance, Lee et al. [START_REF] Lee | Suppression Aeroelastic Instability Using Broadband Passive Targeted Energy Transfers, Part 1: Theory[END_REF] show that a NES coupled to a wing in subsonic flow can partially or even completely suppress flutter instability by TET from the wing to the NES.

The simpler helicopter model that can describe HGR is well known. It involves only lag motion of the four blades and one direction of the fuselage motion. Linear stability of this model shows that HGR is due to a frequency coalescence between a lag mode and the fuselage mode and predicts the range of rotors speeds Ω for which this frequency coalescence occurs. In this work a NES is coupled to the fuselage in an ungrounded configuration (see Fig. 1). The differential equations which govern the time evolution of the fuselage displacement y(t), the NES displacement h(t) and the blades lagging angles δ i (t) (with i ∈ [START_REF] Coleman | Theory of self excited mechanical oscillations of helicopter rotors with theory of self excited mechanical oscillations of helicopter rotors with hinged blades[END_REF][START_REF] Lee | Suppression Aeroelastic Instability Using Broadband Passive Targeted Energy Transfers, Part 1: Theory[END_REF]) can be easily derived using Lagrange's equations. These equations of motion are:

                             (M + 4m δ ) ÿ + c y ẏ + k y y + µ( ẏ -ḣ) + α(y -h) 3 +m δ 4 i=1 L δi cos (ψ i + δ i ) -L Ω + δi 2 sin (ψ i + δ i ) = 0 (1a) m h ḧ + µ( ḣ -ẏ) + α(h -y) 3 = 0. (1b) m δ L 2 δi + c δ δi + k δ δ i + m δ Lÿ cos (ψ i + δ i ) = 0 (1c)
with

ψ i = Ωt -2π i -1 N ,
where N = 4 is number of blades, M is the mass of the fuselage, m δ is the mass of one blade, m h is the mass of the NES (M > m h ), c y , c δ and µ are damping coefficients, k y and k δ are linear stiffness coefficients and α is the cubic stiffness coefficient of the NES.

Results. Numerical parametric investigation of the system of equations ( 1) is performed. It consists to determine the possible regimes in the parameter space. We show that it is possible to partially or even completely suppress HGR by passively transferring energy from the fuselage to the NES. As in [START_REF] Lee | Suppression Aeroelastic Instability Using Broadband Passive Targeted Energy Transfers, Part 1: Theory[END_REF], three responses for the fuselage motion are highlighted: strongly modulated response (SMR), partial suppression and complete suppression of the HGR. An example of SMR is depicted in Fig. 2.

We are currently working on the design of the NES. The aim is to know if the NES parameters leading to a good control of the HGR are compatible with industrial applications.

  Control of helicopter ground resonance by means of passive non-linear energy sink B. Bergeot a , S. Bellizzi b , B. Cochelin c

Acknowledgements. The authors wish to thank Mr. François Malburet for helpful discussions and suggestions. This research is done within the framework of the industrial chair "Dynamique des Systèmes Mécaniques Complexes (Dynamics of complex mechanical systems)" financed by the Airbus Group Foundation.