Michel Perdicakis
email: michel.perdicakis@univ-lorraine.fr

Quan Qin

Maria Bertucci

Hugues Aubriet

Voltammetry of MicroParticles in Thin Layer

Keywords: Voltammetry of microparticles, Manganese dioxide, Ferric oxide, Alumina supported palladium, Callovo-Oxfordian argillite

We describe a new experimental device for the characterization of electroactive microparticles and of the soluble species released during their electrochemical transformation.

By using only a few microparticles, it provides, in a faster way, similar information as modified carbon paste electrodes with electrolytic binders (CPEEBs) and the sensitivity and resolution are must better. The device geometry was modelled with ferrocyanide ions and the voltammograms for ferrocene, pyrolusite and hematite were compared with the literature ones concerning CPEEBs. Finally, an alumina-supported palladium catalyst and pyrite-containing argillite were successfully characterized.

Introduction

Voltammetry of microparticles (VMP) is a technique developed by Scholz [START_REF] Scholz | Voltammetry of solid microparticles immobilized on electrode surfaces[END_REF] that can be applied for the voltammetric characterization of any solid electroactive material, conducting or not, via the mechanical immobilization of microparticles on the cross section of a current collector [START_REF] Scholz | Electrochemistry of Immobilized Particles and Droplets[END_REF][3][4][START_REF] Doménech-Carbó | Sequential identification of organic dyes using the voltammetry of microparticles approach[END_REF]. This latter is most often a graphite rod which had been previously impregnated with solid paraffin to fill up the graphite pores. VMP has the advantages over carbon paste electroactive electrodes of requiring only a few micrograms of matter to be analyzed and avoiding the use of organic binders that can influence the electrochemical behaviour of the solid compounds. However, VMP cannot provide information on the soluble species released in solution by the immobilized microparticles during their electrochemical transformation.

when the electrochemical kinetics is slow and voltammetry at rapid scan rates is inoperable.

To get this information today, electrochemists can utilize Carbon Paste Electrodes with Electrolytic Binders (CPEEBs), which have been developed in the seventies by the Bauer group who established a theoretical model on their working principle [START_REF] Bauer | Etude du comportement de la pâte de carbone à compose électroactif incorporé[END_REF][START_REF] Lamache | Étude, par chronoampérométrie, de l'électrode à pâte de carbone à liant électrolytique et solide électroactif incorporé[END_REF][START_REF] Lamache | Electrode à pate de carbone à liant électrolytique et composes électroactifs peu solubles incorpores: aspects quantitatifs[END_REF]. CPEEBs are prepared by mixing graphite powder with electrolytic solutions instead of non-conducting pasting liquids with traditional modified carbon paste electrodes [START_REF] Kalcher | Chemically modified carbon paste electrodes in voltammetric analysis[END_REF]. In this way, in principle, the whole of the electroactive matter inserted in the carbon paste is electrochemically transformed and soluble species released are trapped in the carbon paste. These electrodes have been successfully used with various compounds just as well insoluble (ferrocene [START_REF] Bauer | Etude du comportement de la pâte de carbone à compose électroactif incorporé[END_REF],

uranium ores [START_REF] Lamache | Etude du comportement électrochimique de sulfures naturels peu solubles : chalcocite, covellite, chalcopyrite[END_REF] copper sulphides [START_REF] Brage | Dosage de sulfures de cuivre à l'aide de l'électrode à pâte de carbone[END_REF][START_REF] Brage | Contribution à l'étude des sulfures de cuivre non stoechiométriques[END_REF][START_REF] Lamache | Anodic oxidation of cuprous sulfide and the preparation of nonstoichiometric copper sulphide[END_REF], ilmenites [START_REF] Andriamanana | Etude électrochimique de différentes ilménites[END_REF], oxides of iron [START_REF] Mouhandess | Influence de la morphologie de l'oxyde de fer,  Fe2O3, sur son comportement électrochimique[END_REF][START_REF] Encinas | Electrochemical study of iron(II) and iron(III) compound mixtures in the solid state. Application to magnetite characterization[END_REF][START_REF] Lecuire | Comportement électrochimique de différents oxydes de fer. I. Etude voltampérométrique de la réduction en milieu acide de poudres cristallisées incluses dans une pâte de graphite[END_REF][START_REF] Lecuire | Comportement électrochimique de différents oxydes de fer. II. Etude des espèces passant en solution par la méthode du disque et de l'anneau[END_REF][START_REF] Lecuire | Réduction électrochimique des oxydes de fer: Application à la mesure de non stoechiométrie[END_REF][START_REF] Bachiller | Electroanalytical study of copper and iron compounds in the solid state: application to copper ferrite characterization[END_REF], copper [START_REF] Bachiller | Electroanalytical study of copper and iron compounds in the solid state: application to copper ferrite characterization[END_REF] and manganese [START_REF] Lorenzo | Electrochemical study of manganese and iron compounds at carbon paste electrodes with electrolytic binder. Application to the characterization of manganese ferrite[END_REF]) as dissolved directly in the electrolyte (FeCl2 and MnCl2 [START_REF] Lorenzo | Electrochemical study of manganese and iron compounds at carbon paste electrodes with electrolytic binder. Application to the characterization of manganese ferrite[END_REF], Fe(CN)6 4-and O2 [START_REF] Bauer | Etude du comportement de la pâte de carbone à compose électroactif incorporé[END_REF]). Ramirez et al. proposed some experimental modifications [START_REF] Ramírez | Carbon paste electrodes with electrolytic binder: Influence of the preparation method[END_REF] and the Vittori group introduced theoretical complements on the Bauer model [START_REF] Mouhandess | Electrochemical behaviour of α-Fe2O3 using carbon paste electrodes: Influence of particle size[END_REF][START_REF] Mouhandess | Some theoretical aspects of electrodissolution of iron oxide (α-Fe2O3) in carbon paste electrodes with acidic binder[END_REF]. Nevertheless, when CPEEBs are employed, extremely slow scan rates (in the range of some hundredths or tenths of mV/s) and very concentrated supporting electrolytes are used in order to minimize capacitive currents. Moreover, the effective deaeration of the carbon paste is quite hard (use of vacuum and glove boxes or previous electrochemical reduction of atmospheric oxygen if possible). In addition, if the solubilization of the non-conducting electroactive microparticles is very slow, the faradaic current becomes of the same order of magnitude as the background current [START_REF] Bauer | Etude du comportement de la pâte de carbone à compose électroactif incorporé[END_REF].

We developed an experimental device that comes from the combination of VMP and thinlayer (TL) voltammetry [START_REF] Reilley | Electrochemistry Using Thin-Layer Cells[END_REF] that we called 'Voltammetry of MicroParticles in Thin Layer' (VMPTL) [START_REF] Perdicakis | Voltammetry of MicroParticles in Thin Layer (VMPTL)[END_REF]. Its principle consists in entrapping a very small amount of electroactive material (some micrometric or submicrometric particles) and a TL of electrolyte solution between the cross sections of two graphite electrodes. VMPTL provides similar information as CPEEBs while maintaining the benefits of VMP. In this way, the voltammetric characterization of both the entrapped microparticles and the soluble species released by the latter can be performed with a much higher sensitivity and resolution than CPEEBs with only a few micrograms of matter. In addition, the solution can be easily deaerated before the entrapping stage by bubbling an inert gas and ten times faster scan rates can be used.

The aim of this paper is to show the potentiality and performance of VMPTL in comparison with CPEEBs. After the description of the experimental device, its functioning will be tested with the model electrochemical system Fe(CN)6 3-/4-. Then, the voltammograms recorded for ferrocene (Fc), pyrolusite (MnO2) and hematite (Fe2O3) will be compared with the literature ones obtained with CPEEBs. Finally, VPMTL will be utilised for the characterization of two electroactive compounds that are at percent levels in more or less complex matrixes: metallic palladium in an alumina-supported catalyst and pyrite (FeS2) in Callovo-Oxfordian argillite.

The present approach could be also useful for studies of organic deposits, as it has been pioneered by Doménech-Carbó [START_REF] Doménech-Carbó | Electrochemical tomato (Solanum lycopersicum L.) characterisation using contact probe in situ voltammetry[END_REF][START_REF] Doménech-Carbó | Electrochemistry-based chemotaxonomy in plants using the voltammetry of microparticles methodology[END_REF].

Experimental

The electrochemical cell

Experimental setup: some electroactive microparticles and a TL of solution are entrapped between the cross sections of two graphite rods. The rods are mounted, with an epoxy resin (Epofix kit from Struers), in the axis of the truncated plungers of two glass syringes that allow to the rods to slide hermetically inside the end truncated syringe barrels (Fig. 1A). The rods are externally short-circuited and connected to the working electrode input of the potentiostat.

The main purpose of using two short-circuited graphite electrodes opposite each other instead of only one against an insulating surface, as with classic thin layer cells, is to ensure a good electrical contact with the immobilized electroactive microparticles allowing the electrochemical transformation of the greatest possible number, if not all, of them.

Before mounting the graphite rods (3.05 mm diameter, 99.9995% metals basis from Alfa Aesar) in the truncated plungers of the syringes it is necessary to fill up the graphite pores with solid paraffin to eliminate the electrochemical response of adsorbed oxygen and to avoid the contamination of graphite by soluble species as recommended by Scholz [START_REF] Scholz | Voltammetry of solid microparticles immobilized on electrode surfaces[END_REF].

Cell design: the cell, depicted schematically in Figure 1B, is made of Perspex and has a cylindrical shape. There is a central cylindrical conduit, containing the electrolytic medium, where are stuck horizontally with a bicomponent epoxy glue the syringe barrels where the two graphite electrodes are allowed to move. A platinum counter electrode inserted in a PTFE stopper is screwed on the basis of the cell and, according to the experiment, a silver-wire used as the quasi-reference electrode or a Ag/AgCl reference electrode are inserted in a PTFE stopper on the top of the cell. An inert gas inlet tube is also inserted in the top stopper.

Depending on the position of the tube, the gas is bubbling through the solution or it maintains an inert atmosphere over its surface. The device is equipped with a metallic holder and a slight rotation about its axis allows to remove one after the other the graphite electrodes for cleaning or polishing without emptying the cell.

Use of the device

To transfer and immobilize the electroactive particles inside the TL, a few grains are applied to the graphite surface of one of the electrodes and pressed on it with a small spatula. Then the electrode is softly struck against a glass surface before its introduction in the electrochemical cell in order to remove the particles that were badly fixed. After deaeration of the solution, the graphite electrodes are pushed up and rotated one against the other in such a way that the immobilized microparticles come in the most intimate contact with the two graphite discs.

Before each experiment the surface of the electrodes is cleaned by polishing with a filter paper or with a grinding paper, if necessary.

Electrochemical measurements

Voltammetric experiments were performed with a model 174A polarographic analyzer (EG&G-PAR), which was interfaced via a computer AD card and monitored by a home-made application created by means of the TestPoint TM software, or with a 'Quadstat' four-channel potentiostat used as a single three-electrode potentiostat and monitored by the EChem Software (both from EDaq). According to the experiments, the potentials were referred to the silver quasi-reference electrode (Ag/QREF) or to the saturated Ag/AgCl electrode.

Chemicals and minerals

Potassium hexacyanoferrate(II), sulphuric and hydrochloric acids and potassium chloride, were of RP Normapur grade from Prolabo. Ferrocene purum grade, manganese(IV) oxide puriss. grade and iron(III) oxide puriss. p.a. grade were obtained from Fluka, and palladium(II) chloride from Merck. Palladium on alumina pellets loading 5 wt. % catalyst was from Johnson Matthey (see SECM characterization in Fig. 2A). The Callovo-Oxfordian argillite sample containing 0.5-1 wt. % pyrite [START_REF] Gaucher | ANDRA underground research laboratory: interpretation of the mineralogical and geochemical data acquired in the Callovian-Oxfordian formation by investigative drilling[END_REF] came from Bure (Meuse) in France (optical microscope picture in Fig. 2B). It was kindly provided by Prof. M. Cathelineau ('GeoRessources' laboratory -Université de Lorraine).

Results and Discussion

Test of the device with the Fe(CN)6 3-/Fe(CN)6 4-system

A series of cyclic voltammograms obtained for 2 mM K4Fe(CN)6 in 1 M KCl using scan rates from 1 to 5000 mV/s allowed us to estimate the liquid TL thickness. Some typical voltammograms recorded in the 10-100 mV/s range are displayed in Figure 3A. The analysis of the results shows that the anodic and cathodic peak separation (Ep) first decreases until the scan rate (v) is over 6 mV/s then increases (Fig. 3B). Ep remains smaller than 58 mV as long as v is lower than 40 mV/s. The plot of the anodic peak current (Ip) versus v in a log-log representation (Fig. 3C) exhibits two linear parts with an interconnecting break point at about 100 mV/s. The slope of the first part, very close to 1, indicates a TL behaviour, [START_REF] Scholz | Voltammetry of solid microparticles immobilized on electrode surfaces[END_REF]. The second part has a slope very close to 0.5 according to the cyclic voltammetry theory: [START_REF] Scholz | Electrochemistry of Immobilized Particles and Droplets[END_REF]. In these formulas, C° and D are, respectively, the concentration and the diffusion coefficient of the electroactive species, V the volume of the TL and A the surface area of the electrode. The variation of the slope of log Ip vs. log v is interpreted as follows: As long as the diffusion layer thickness remains wider than the distance between the graphite electrodes one observe a TL behaviour.

Ip = [n 2 F 2 VC°/(4RT)]v
Ip = [0.4463(F 3 /[RT]) 1/2 n 3/2 AD 1/2 C°]v 1/2
Then, when v becomes high enough for the diffusion layer to become smaller than the TL, the system varies with v according to the cyclic voltammetry theory. The scan rate value at the break point (vbp) depends only on the TL thickness (l) and D. In fact, at this point, Ip can be calculated by both equations ( 1) and ( 2) and one can easily demonstrate that l = 1.7852[DRT/(nFvbp)] 1/2 (3). According to the experimental data, vbp = 96 mV/s that gives 24.6 µm for l. As the VMPTL device comprises two parallel active graphite electrodes, l must be multiplied by 2 to have the real interelectrode distance, therefore, the distance is about 50 µm.

Figure 4C also shows the modelling made using equations ( 1) and ( 2) with D Fe(CN)6 4-= 7.410 -6 cm²/s [START_REF] Trémillon | Electrochimie analytique et réactions en solution[END_REF] for two 3.0 mm diameter electrodes put in front each other and forming two 24.6 µm TLs; the calculated curve is in good agreement with the experimental points.

Comparison with literature experiments performed with CPEEBs

In this section the voltammograms recorded for ferrocene, pyrolusite and hematite using VMPTL are compared with those obtained by Bauer et al. [START_REF] Bauer | Etude du comportement de la pâte de carbone à compose électroactif incorporé[END_REF] and Lorenzo et al. [START_REF] Bachiller | Electroanalytical study of copper and iron compounds in the solid state: application to copper ferrite characterization[END_REF][START_REF] Lorenzo | Electrochemical study of manganese and iron compounds at carbon paste electrodes with electrolytic binder. Application to the characterization of manganese ferrite[END_REF] with CPEEBs. Four i-E curves from these references have been digitized and superimposed on our own experimental data to aid comparison.

Characterization of ferrocene

Ferrocene has been used in the very first experiments performed with CPEEBs in the seventies by the Bauer group. Voltammograms for ferrocene microparticles recorded in 1.4 M H2SO4 using VMPTL, at 2 mV/s, and a CPEEB at 0.2 mV/s (from ref. [START_REF] Bauer | Etude du comportement de la pâte de carbone à compose électroactif incorporé[END_REF]) are shown in Figure 4A. Overall, the shape of the two curves are quite comparable in spite of a scan rate 10 times faster for VMPTL. At the first anodic scan, started from the open circuit potential (OCP), the oxidation of ferrocene is spread over several hundreds of millivolts because of the ohmic drop resulting from the bad contact between the insulating ferrocene particles and graphite. In contrast, the reduction of the produced soluble ferrocenium cation gives rise to the symmetrical cathodic peak C1 at potentials close to the thermodynamic value. The second anodic scan is much less affected by ohmic drop because of the intimate contact between graphite and the ferrocene deposited on it during the reduction of Fc + at C1. However, in the case of CPEEB, peak A1 is far from being symmetrical because of remaining ohmic drop. Some typical cyclic voltammograms recorded with the VMPTL device after stabilization of the currents are displayed in Figure 4B. Thus, in the case of ferrocene, VMPTL provides more rapidly similar information to the one obtained using CPEEBs and the corresponding voltammograms are less affected by ohmic drop.

Characterization of pyrolusite

The voltammogram recorded with MnO2 microparticles in 10 mM HCl using VMPTL at 5 mV/s and that displayed in ref. [START_REF] Lorenzo | Electrochemical study of manganese and iron compounds at carbon paste electrodes with electrolytic binder. Application to the characterization of manganese ferrite[END_REF] (CPEEB, v = 0.5 mV/s) are represented in Figure 5. The scans were started from the OCP in the negative direction. Both voltammograms provide similar information, however, the voltammogram recorded using VMPTL does not suffer neither from ohmic drop nor from capacitive current as that using CPEEB, in spite of a scan rate ten times faster.

According to Lorenzo et al. [START_REF] Lorenzo | Electrochemical study of manganese and iron compounds at carbon paste electrodes with electrolytic binder. Application to the characterization of manganese ferrite[END_REF], the cathodic peak C1°, observed at the first cycle, corresponds to the reduction of the initial MnO2 in soluble Mn 2+ ions (MnO2 + 4H + + 2e - Mn 2+ + 2H2O). After the scan reversal, the opposite reaction occurs at anodic peak A1, which is located just before the anodic limit. After another reversal of the sweep direction, the cathodic signal for the reduction of MnO2 electrochemically deposited on the graphite surface (peak C1) is shifted towards more positive potentials with respect to peak C1°. A careful examination of the voltammogram obtained with VMPTL shows that: (i) The C1/C1° peaks area ratio is about 1.1 (This ratio is approximately 2.6 in the case of the CPEEB because of the presence of Mn 2+ ions in the electrolyte inside the carbon paste. According to the authors, Mn 2+ result from the interaction of MnO2 with HCl, however, the corresponding chemical reaction is not written and the process is unclear for us). (ii) In the VMPTL voltammogram, a very slight signal is still present at C1° during the second cycle that means that the MnO2 reduction is not totally complete. When a CPEEB is used, it is very difficult to estimate the amount of MnO2 reduction at the second scan because of the probable overlapping of peaks C1° and C1.

Characterization of hematite

The voltammogram for the characterization of Fe2O3 in 1M KCl at pH 1 using VMPTL is shown in Figure 6A (curve a). The potential scan (v = 5 mV/s) started from the OCP in the negative direction. The cathodic peak C1° corresponds to the Fe2O3 reduction into Fe 2+ that is oxidized to Fe 3+ during the reverse scan (peak A1). Along the second and next cycles Fe 3+ is reduced in Fe 2+ (C1) and then oxidized again in Fe 3+ (A1). This voltammogram closely resembles that obtained by Encinas Bachiller et al. [START_REF] Bachiller | Electroanalytical study of copper and iron compounds in the solid state: application to copper ferrite characterization[END_REF] in 1M HCl by means of a CPEEB at 0.5 mV/s. However, in the case of VMPTL, because of the lower acidity of the solution, the reduction of Fe2O3 takes place at more negative potentials and the peak shape for the Fe 3+ / 2+ system is different. Indeed, peaks A1 and C1 related do CPEEB demonstrates pure TL behaviour but are not symmetrical because of the ohmic drop, whereas those related to VMPTL are almost symmetrical. Moreover, even if peaks A1 on both voltammograms are virtually superimposable, the A1-C1 peak separation is larger in the case of VMPTL because of the formation of FeOH 2+ and Fe(OH)2 2+ species whose reduction is more difficult than that of Fe 3+ [START_REF] Azra Bilgin | Effects of Soluble Ferri-Hydroxide Complexes on Microbial Neutralization of Acid Mine Drainage[END_REF].

Figure 6B illustrates the comparison of the voltammograms for Fe2O3 recorded with VMPTL (v = 5 mV/s) and CPEEB (v = 0.5 mV/s, from ref. [START_REF] Lorenzo | Electrochemical study of manganese and iron compounds at carbon paste electrodes with electrolytic binder. Application to the characterization of manganese ferrite[END_REF]) in the same medium: 10 mM HCl. As in the case of MnO2, the voltammogram corresponding to VMPTL shows no ohmic drop nor capacitive current if compared to that for CPEEB. However, the two voltammograms are completely different. The one for CPEEB provides similar information as provided in the 1M HCl medium. When VMPTL is used, the Fe2O3 reduction is far to be complete, the signal A1 for Fe 2+ oxidation is flat while that for Fe 3+ reduction (C1) is barely visible. This due to the very low quantity of protons inside the TL. Indeed, the electrochemical reduction of hematite according to the reaction: Fe2O3 + 2e -+ 6H +  2Fe 2+ + 3H2O (1) requires 2 electrons and 6 protons.

Indeed, assuming that in the best-case scenario the TL thickness is 100 µm, the corresponding TL volume is 710 -7 L, which contains 710 -9 mol of protons. By integration of the voltammetric curve in Figure 6B, we estimated at approximately 4.310 -4 C the charge exchanged at peak C1° and, consequently, the reduction concerns 2.210 -9 mol of Fe2O3 that require 1.310 -8 mol of H + . Yet, only 710 -9 mol are present in the TL. In this way, the pH inside the TL will be highly alkaline (12.3) after the hematite reduction, which explains the Fe 3+/2+ system shape. We also made the same kind of calculations for the voltammogram obtained with the CPEEB [START_REF] Lorenzo | Electrochemical study of manganese and iron compounds at carbon paste electrodes with electrolytic binder. Application to the characterization of manganese ferrite[END_REF]. Given that 1.6 mg of Fe2O3 (10 -5 mol) and 50 µL of 10 -2 M HCl (510 -7 mol) were used for the preparation of the carbon paste there is a serious shortage of protons for the complete reduction of hematite. We estimated that approximately 710 -7 mol of Fe2O3 were reduced at peak C1°, by integration of the voltammetric curve, which correspond to a 0.14 C charge. It is unclear in ref. [START_REF] Lorenzo | Electrochemical study of manganese and iron compounds at carbon paste electrodes with electrolytic binder. Application to the characterization of manganese ferrite[END_REF] if the electrode was filled with the whole of the carbon paste or not. We accepted the most favourable hypothesis i.e. the totality of the carbon past was used and therefore 510 -7 mol of H + are present inside the electrode, whereas only 7% of the initial hematite was electrochemically reduced. As the reduction of 710 -7 mol of Fe2O3 requires 4.210 -6 mol of H + , the H + /Fe2O3 ratio is much more unfavourable than in the case of VMPTL and the shape of the voltammogram in Figure 8B is totally incompatible with such a high pH value unless there is a massive introduction of H + from the solution into the paste.

Figure 6C illustrates an attempt to prevent the Fe(III) precipitation in the 10 mM HCl medium by formation of the anionic complex FeCl4 -after the addition of a massive amount of chloride ion. This is why the series of voltammograms in Figure 6C was recorded in the 10 mM HCl + 1 M KCl mixture by decreasing dramatically the amount of Fe2O3 immobilized in the TL. In the forward scan, the signal C1° for the Fe2O3 reduction is no more a sharp peak, as in Figure 6B, but a wide signal that is composed of two successive steps at -0.20 and -0.37 V as in Figure 6A. After the reversal of the scan direction, a well defined anodic peak A1 is observed at 0.48 V for the Fe(II) oxidation, then, following a second scan reversal, peak C1 for the Fe(III) reduction appears at 0.42 V. However, the C1/A1 height ratio is only 0.69. During the next cycles, the cathodic signal C1° remains virtually unchanged whereas peaks A1 and C1 progressively grow towards constant heights with an A1/C1 ratio about 0.77. What is notable is that it is as hematite was regenerated at every voltammetric cycle. Probably, the mechanism is complex and may involve the formation of green rusts [START_REF] Génin | Preparation and Eh-pH diagrams of Fe(II)-Fe(III) green rust compounds; hyperfine interaction characteristics and stoichiometry of hydroxy-chloride, -sulphate and -carbonate[END_REF]; therefore, specific investigation is required that is beyond the scope of this article.

Characterization of electroactive compounds present at percent levels in different matrices

After showing the interest of VMPTL for the characterization of pure solids, it is extended here in species that are present at very low amounts in more or less complex matrices, i.e. a palladium-based catalyst and a pyrite containing natural rock.

Palladium supported on alumina catalyst (Pd/Al/Cat)

The hydrogenation aromatic rings, the oxidation of methane [START_REF] Simone | Reversible poisoning of palladium catalysts for methane oxidation[END_REF] and glucose [START_REF] Nikov | Palladium on alumina catalyst for glucose oxidation: reaction kinetics and catalyst deactivation[END_REF], the deoxygenation of inert gases [START_REF] Nowicki | Method for preparing catalysts for oxygen removal from neutral gases[END_REF], the fuel cells [START_REF] Wang | Hydrogen production on Ni-Pd-Ce/γ-Al2O3 catalyst by partial oxidation and steam reforming of hydrocarbons for potential application in fuel cells[END_REF][START_REF] Weston | Combination of zeolite and alumina impregnated with noble metal(s) for carbonyl sulfide and the removal at low temperatures in fuel cell processor applications[END_REF] and the combustion in gas turbines [START_REF] Vatcha | Low-emission gas turbines using catalytic combustion[END_REF] are among the numerous applications of Pd/Al/Cats. In the environmental area, Pd/Al/Cats are used for the reductive dehalogenation of chlorinated hydrocarbons or the hydrogenation of aromatic compounds in groundwater [START_REF] David | Control of catalytic debenzylation and dehalogenation reactions during liquid-phase reduction by H2[END_REF][START_REF] Siantar | Treatment of 1,2-dibromo-3chloropropane and nitrate-contaminated water with zero-valent iron or hydrogen/palladium catalysts[END_REF] and the exhaust emissions (NOx+CO) transformation in the automotive [START_REF] Gonzalez-Velasco | HREM and XRD characterization of thermal ageing of Pd/CeO2/Al2O3 automotive catalysts[END_REF][START_REF] Muraki | Nitric oxide reduction performance of automotive palladium catalysts[END_REF]. Vibrational spectroscopy has been mainly used for the Pd/Al/Cats characterization [START_REF] Mamede | In situ Raman characterisation of surface modifications during NO transformation over automotive Pd-based exhaust catalysts[END_REF][START_REF] Mamede | Surface Raman spectroscopic study of NO transformation over Pd-based catalysts[END_REF][START_REF] Le Bourdon | In situ characterization by Raman and IR vibrational spectroscopies on a single instrument: DeNOx reaction over a Pd/γ-Al2O3 catalyst[END_REF][START_REF] Horiba | Situ Characterisation of Heterogeneous Catalytic Reactions by Raman and IR Vibrational Spectroscopies on a single Instrument[END_REF]; it showed that an oxidation phenomenon occurred on the Pd surface with formation of PdO when the catalysts were in contact with air, even at room temperature. Their regeneration requires the reduction of the oxide by gaseous hydrogen at high temperatures. Finally, the Bauer group investigated in 1992 the electrochemical characterization of palladium compounds, including palladium automotive catalysts, by means of CPEEBs [START_REF] Adekola | Electrochemical study of some palladium compounds at a carbon paste electrode-application to the determination of palladium in oxidation automotive catalysts[END_REF][START_REF] Adekola | Electrochemical behaviour of some automotive catalysts at a carbon paste electrode and with electrolytic binder[END_REF].

The experiments were performed in 1 M KCl at pH 2, in order to lower the Pd° oxidation potential by the formation of palladium chloro-complexes and avoid the production of palladium monoxide which is insoluble in all acids, including aqua regia [START_REF] Remy | Treatise on Inorganic Chemistry[END_REF][START_REF] Duval | Nouveau Traité de Chimie Minérale[END_REF]. The speciation diagram that we plotted for 50 mM Pd(II) in 1 M KCl shows that if pH<2.25 the whole quantity of Pd(II) is under the anionic forms PdCl4 2-(95%) and PdCl3 -(5%).

The VMPTL voltammograms recorded at 5 mV/s for Pd/Al/Cat microparticles obtained after grinding of the commercial pellets are displayed in Figures 9A and9B.

In Figure 9A, the potential sweep was started from the OCP in the positive direction. At the first voltammetric cycle the anodic peak A1° (0.64 V) corresponds to the oxidation of the alumina-supported metallic Pd particles in PdCl4 2-. After the potential reversal, the cathodic peak C1° (0.00 V) is ascribed to the deposition on the graphite surface of Pd° that results from the reduction of: (i) PdCl4 2-, released at A1° and (ii) PdO originally present on the catalyst surface (see Fig. 9B). Indeed, the peak C1° area is much greater than that of A1° (C1°/A1°1.7). At the second cycle, the stripping of Pd° deposited on the graphite surface during C1° occurs at lower potentials (peak A1 at 0.58 V) than that originally present on Al2O3. However, the A1/C1° area ratio is about 0.6; that means that either a large part of the Pd° formed at C1° is no more in electric contact with the graphite or the air-oxidized palladium on the Pd/Al/Cat surface was at an oxidation state higher than 2. After the scan direction reversal, PdCl4 2-is reduced at peak C1 (0.02 V), which is thinner than C1° because of the absence of PdO, and C1/A1  1.2. At the third and forth cycles, the C1/A1 area ratio tends towards 1 and the voltammetric peaks height progressively decreases. This is probably the result of the gradual adsorption of PdCl4 2-on alumina, which, according to electrokinetic measurements [START_REF] Kosmulski | pH-dependent surface charging and points of zero charge II. Update[END_REF], is positively charged for pH values smaller than 8-9 whatever the origin and the crystalline structure of Al2O3. The positions of peaks C1 (0.02 V), for the PdCl4 2- reduction, and A1 (0.58 V) for the Pd° stripping on graphite have been confirmed by the voltammogram recorded for 1 mM PdCl2 in the same medium (Fig. 9C curve c). Therefore, the kinetics of the PdCl4 2-/Pd system is slow. In the literature the positions of peaks A1 and C1 are very dependent on the experimental conditions: 0.66 and 0.00 V vs. SCE, respectively, at a carbon felt in 0.1 M HCl [START_REF] Polcaro | Electrodeposition of catalysts for hydrogenation of organic molecules: hydrogenation of benzaldehyde[END_REF], 0.36 and 0.174 V vs. SCE at a CPEEB in 2 M HCl [START_REF] Adekola | Electrochemical study of some palladium compounds at a carbon paste electrode-application to the determination of palladium in oxidation automotive catalysts[END_REF], 0.67 and 0.26 V vs. SCE at a vitreous carbon electrode in 0.03 M HCl + 0.8 M HClO4 [START_REF] Bell | The deposition of palladium[END_REF].

As regards the PdCl4 2-/Pd standard potential, there are some inconsistencies in the literature where E° varies from 0.580 to 0.646 V vs. SHE [START_REF] Golodov | Physicalchemical properties of acido complexes of palladium(II) in aqueous dioxane media[END_REF][START_REF] Fasman | Reactivity of complex compounds of palladium(II) in aqueous solutions[END_REF][START_REF] Vanýsek | CRC Handbook of Chemistry and Physics[END_REF][START_REF] Nikolaeva | Effect of temperature on standard potentials of palladium(II) halo complexes[END_REF][START_REF] Polotnyanko | Thermodynamic properties of Pd chloride complexes and the Pd 2+ (aq) ion in aqueous solutions[END_REF][START_REF] Templeton | The Formal Electrode Potentials of Palladium in Aqueous Hydrochloric and Perchloric Acid Solutions. Stability of Chloropalladite Ion[END_REF][START_REF] Tagirov | The speciation and transport of palladium in hydrothermal fluids: Experimental modeling and thermodynamic constraints[END_REF]. The PdO reduction potential, which depends on the pH, was observed at -0.165 vs. Ag/QREF at pH 2 in the present study, -0.100 vs SCE at a CPEEB in 2M HCl [START_REF] Adekola | Electrochemical study of some palladium compounds at a carbon paste electrode-application to the determination of palladium in oxidation automotive catalysts[END_REF], 0.529 V vs. the reversible hydrogen electrode (RHE) at a Pd electrode in 0.5 M H2SO4 [START_REF] Dall'antonia | Influence of temperature on the growth of surface oxides on palladium electrodes[END_REF] and 0.76 vs. RHE at a smooth Pd electrode in 1M H2SO4 [START_REF] Burke | An electrochemical investigation of monolayer and multilayer oxide films on palladium in aqueous media[END_REF]. The E°P dO/Pd standard potential, ranges between 0.87 and 0.91 V vs. SHE [START_REF] Duval | Nouveau Traité de Chimie Minérale[END_REF][START_REF] Golodov | Physicalchemical properties of acido complexes of palladium(II) in aqueous dioxane media[END_REF][START_REF] Dall'antonia | Influence of temperature on the growth of surface oxides on palladium electrodes[END_REF][START_REF]Techniques de l'ingénieur, Archives, Enthalpie et Enthalpie libre de Formation[END_REF][START_REF] Pourbaix | Electrochemical Properties of the Platinum Metals[END_REF]. Therefore, the normal apparent potential at pH 2 is about 0.80 V vs. SHE (0.57 V vs. Ag/QREF).

When the voltammograms are recorded in the negative direction (Fig. 9B), at the first cycle, the broad signal C0 going from 0.1 to -0.2 V corresponds to the PdO reduction in Pd°. After reversal of the scan direction, two anodic stripping signals appear: (i) A0 for the elemental palladium deposited onto graphite surface during the PdO reduction and (ii) A1° to the Pd originally present on Al2O3. The (A0+A1°)/C0 peaks area ratio is about 3. Then, after the scan reversal (A0+A1°)/C1 = 1.3. At the second and third cycles, as for the voltammograms recorded in the positive direction, the heights of peaks A1 and C1 progressively decreases and their ratio tends to 1.

The voltammogram obtained with an automotive Pd catalyst using a CPEEB at a 0.25 mV/s scan rate in 2 M HCl (digitized from reference [START_REF] Adekola | Electrochemical study of some palladium compounds at a carbon paste electrode-application to the determination of palladium in oxidation automotive catalysts[END_REF]) is represented in Figure 9C (curves a andb). The comparison with the VMPTL voltammograms shows that, even if the potentials of peaks A0, A1, C0 and C1 are different because of the high acidic level used with the CPEEB the interpretation is the same. Thus, the results are qualitatively similar and those for VMPTL can be obtained 20 times faster.

Pyrite containing Callovo-Oxfordian argillite (COx) from Bure

In several studies on the underground landfill of spent nuclear fuel, it is accepted that small amounts of pyrite (~1 wt. %) will be responsible for the low redox potential of argillaceous soils envisaged for the fuel storage [START_REF] Pearson | Mont Terri Project -Geochemistry of Water in the Opalinus Clay Formation at the Mont Terri Rock Laboratory[END_REF]. Therefore, it is important to examine if electrochemical techniques could be a useful tool to investigate the reactivity of minerals included in these subsoils. Bure is the French site planned for the deep repository of high activity long life radioactive waste [START_REF]The Meuse/Haute-Marne Underground Research Laboratory (MHM URL): a scientific tool[END_REF]. The experiments have been performed in chloride medium (10 mM HCl + 1 M KCl), because the chloride concentration in the Bure clay poral water ranges from 0.01 to 0.5 M according to the localization of the analysed samples [68].

The voltammogram for COx grains entrapped in the VPMTL device is shown in Figure 10.

The potential scan was started from the OCP (0.07 V vs. Ag/QREF) in the positive direction.

The anodic peak A1 that appears at about 0.99 V, just before the anodic limit, is due to the pyrite oxidation. During the reverse scan, we can see the reduction of species issued from the FeS2 oxidation: Fe(III) at peak C2 (0.40 V), and a mixture of S8, Sx 2-and oxysulphur compounds (S2O3 2-, S4O6 2-…) [START_REF] Kelsall | Electrochemical oxidation of pyrite (FeS2) in aqueous electrolytes[END_REF][START_REF] Antonijević | Influence of inorganic anions on electrochemical behaviour of pyrite[END_REF][START_REF] Descostes | Pyrite dissolution in acidic media[END_REF] at peak C3 (-0.50 V). In the second cycle, one can remark the appearance of peak A2 (0.59 V) for the oxidation of Fe 2+ ions formed at C2 and the dramatic decrease of peak A1. Therefore, almost the whole of pyrite comprised in the clay has been oxidized during the first cycle. Peak C2 is superimposable to that of the first cycle. It is difficult to have an exact estimation of the A1/C2 peak area ratio because A1 and the oxidation of chloride ions are too close. However, the very approximate experimental value of 16 that we determined is in favour of the formation of oxysulphur compounds like SO4 2-, S2O3 2-, S4O6 2-…during the oxidation of pyrite. For example, the reaction FeS2 + 8 H2O + Cl -  FeCl 2+ + 2 SO4 2-+ 16 H + + 15 e -would lead to an A1/C2 peak area ratio of 15. Because of the fundamental importance of the electrochemical characterization of COx, a more detailed presentation will be the subject of a separate paper.

Conclusions

We hope that this article could convince the reader of the usefulness of VMPTL that enables to characterize electrochemically not only pure solids but also compounds that are in very low amounts in complex matrixes. As this technique provides information on the soluble species formed at the surface of the immobilized electroactive microparticles, it is an efficient tool, complementary to VMP. Moreover, VMPTL, from just a few µg of matter, gives equivalent results as CPEEBs using scan rates more than ten times faster with better resolution and higher signal-to-noise ratio, especially if the experiments are made in low conducting media.

The voltammograms recorded in this study with ferrocene, pyrolusite and an aluminasupported palladium catalyst are in good agreement with the literature ones obtained using CPEEBs, while those for hematite lead us to suspect a complex mechanism. Finally, VPMTL has been successfully applied to a sample of Callovo-Oxfordian argillite. and a CPEEB (reproduced with permission from ref. [START_REF] Lorenzo | Electrochemical study of manganese and iron compounds at carbon paste electrodes with electrolytic binder. Application to the characterization of manganese ferrite[END_REF]). Scans started from the OCP in the negative direction.

Figure Captions

Fig. 1 .

 1 Fig. 1. The VMPTL electrochemical cell: (A) experimental setup, (B) schematic

Fig. 2 .

 2 Fig. 2. Electroactive solids present at percent levels in different matrices: (A) Alumina-

Fig. 3 .

 3 Fig. 3. Voltammetric behaviour of 2 mM K4Fe(CN)6 in 1 M KCl using the VMPTL device.

Fig. 4 .

 4 Fig. 4. Voltammetric characterization of ferrocene microparticles in 1.4 M H2SO4 using

Fig. 5 .

 5 Fig. 5. Voltammetric characterization of MnO2 microparticles in 10 mM HCl using VMPTL

Fig. 6 .

 6 Fig. 6. Voltammetric characterization of Fe2O3 microparticles: (A) Curve (a) 1M KCl, pH 1

Fig. 7 .

 7 Fig. 7. Voltammetric characterization of palladium supported on alumina catalyst (all scans

Fig. 8 .

 8 Fig. 8. Voltammetric characterization of Callovo-Oxfordian argillite in 1 M KCl pH 2 using

Fig. 6

 6 Fig. 6

  Fig. 7

Fig. 8

 8 Fig. 8

ISE membre

Acknowledgements

The authors would like to acknowledge Mr. J.P. Moulin for his contribution in the development of the VMPTL device and Prof. Cathelineau for providing the argillite sample.


Giuseppe Pennisi
email: giuseppe.pennisi@onera.fr

Cyrille Stephan
email: cyrille.stephan@onera.fr

Etienne Gourc
email: gourc@insa-toulouse.fr

Guilhem Michon
email: guilhem.michon@isae.fr

Experimental Investigation and Analytical Description of a Vibro-Impact NES Coupled to a Single-Degree-of-Freedom Linear Oscillator Harmonically Forced

In this paper the dynamics of a system composed of a harmonically forced single-degree-of-freedom linear oscillator coupled to a Vibro-Impact Nonlinear Energy Sink (VI-NES) is experimentally investigated. The mass ratio between the VI-NES and the primary system is about 1%. Depending on the external force's amplitude and frequency, either a Strongly Modulated Response (SMR) or a constant amplitude response (CAR) is observed. In both cases an irreversible transfer of energy occurs from the linear oscillator towards the VI-NES: process known in literature as passive

Targeted Energy Transfer (TET). Furthermore, the problem is analytically studied by using the method of multiple scales. The obtaineSlow Invariant Manifold (SIM) shows the existence of a stable and of an unstable branch of solutions, as well as of an energy threshold (a saddle-node bifurcation) for the solutions to appear. Subsequently the fixed points of the problem are calculated. When a stable fixed point is reached, the system is naturally drawn to it and a CAR is established, whereas when no stable point is attained, the system exhibits a SMR regime. Finally a good correlation between the experimental and the analytical results is presented.

Introduction

A Nonlinear Energy Sink (NES) is defined as a vibration absorber with a relatively light mass, nonlinearly attached to a primary system whose vibrations are to be mitigated. The use of a NES as a vibration absorber has been object of interest in the field of nonlinear dynamics in the last decade. Studies have shown that, if compared to the classical linear Tuned Mass Damper (TMD), it could be efficient in a broader frequency range and for a smaller addition of mass to the primary system [START_REF] Gendelman | Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators part ii resonance capture[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators i: dynamics of the underlying hamiltonian system[END_REF]. It has been shown that the nonlinear attachments can lead to an irreversible energy transfer from the primary system towards the NES, this process is known as Targeted Energy Transfer (TET) or energy pumping [START_REF] Manevitch | Parameters optimization for energy pumping in strongly nonhomogeneous 2 dof system[END_REF][START_REF] Kerschen | Irreversible passive energy transfer in coupled oscillators with essential nonlinearity[END_REF].

Experimental works have investigated this phenomenon and are presented in [START_REF] Mcfarland | Experimental study of nonlinear energy pumping occurring at a single fast frequencyexperimental study of non-linear energy pumping occurring at a single fast frequency[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF][START_REF] Kerschen | Theoretical and experimental study of multimodal targeted energy transfer in a system of coupled oscillators[END_REF]. They have shown that the dynamics which governs the energy transfer phenomenon is a 1:1 resonance capture between the primary system and the NES. An important feature of this kind of devices is that they do not have a preferrable frequency of oscillation because of their intrinsic nonlinear nature. Thanks to this characteristic, they are able to tune themselves to the primary system's frequency. TET under external forcing has been investigated theoretically [START_REF] Starosvetsky | Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry[END_REF] and experimentally [START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF] showing that in addition to the steadystate-constant-amplitude-response regime, another type of response can arise which is referred to as Strongly Modulated Response (SMR). NESs have also been studied when applied to passive control of instabilities. In [START_REF] Gendelman | Bifurcations of self-excitation regimes in a van der pol oscillator with a nonlinear energy sink[END_REF] a NES was used to control the limit cycles of a Van der Pol oscillator. In [START_REF] Lee | Suppressing aeroelastic instability using broadband passive targeted energy transfers, part 1: Theory[END_REF][START_REF] Lee | Suppressing aeroelastic instability using broadband passive targeted energy transfers, part 2: Experiments[END_REF][START_REF] Gendelman | Asymptotic analysis of passive nonlinear suppression of aeroelastic instabilities of a rigid wing in subsonic flow[END_REF][START_REF] Vaurigaud | Passive control of aeroelastic instability in a long span bridge model prone to coupled flutter using targeted energy transfer[END_REF] it was used to suppress aeroelastic instabilities.

Most of the works mentioned so far have dealt with a nonlinearity represented by a cubic stiffness. The principle consists in using the geometric nonlinearity of an elastic element to attain a restoring force proportional to the cube of the displacement. Nevertheless, the nature of the nonlinearity may theoretically be of any kind. Later studies have explored other ideas such as: non-polynomial functions [START_REF] Gendelman | Targeted energy transfer in systems with non-polynomial nonlinearity[END_REF], multiple states of equilibrium [START_REF] Gendelman | Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium[END_REF], non-smooth functions and Vibro-Impacts [START_REF] Georgiadis | Shock isolation through passive energy pumping caused by non-smooth nonlinearities[END_REF][START_REF] Karayannis | Vibro-impact attachments as shock absorbers[END_REF][START_REF] Lee | Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments[END_REF][START_REF] Nucera | Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Part ii. experimental results[END_REF]. Many of the first works on Vibro-Impacts were based on numerical simulations. Recently, similarly to what was used for smooth nonlinearities, an analytical approach adopting the multiple-scale analysis has been proposed in [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF] for impulsive forces, and extended to the case of a harmonic forcing in [START_REF] Gendelman | Dynamics of forced system with vibroimpact energy sink[END_REF]. In [START_REF] Gourc | Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink[END_REF] preliminary experimental observations of the different regimes of response has been achieved for a VI-NES applied to a harmonically forced linear oscillator.

In this paper the experimental and the analytical study of the response of a system composed of a Vibro-Impact NES coupled to a single-degree-of-freedom linear oscillator is carried out. The main purpose of this work is to bridge the theoretical and experimental studies, in order to reach a comparison between the analytical results and the experimental observations. The study carried out in [START_REF] Gourc | Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink[END_REF] is used as an important starting point to broaden the general understanding of the VI-NES. In [START_REF] Gourc | Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink[END_REF], experimental investigations have shown how such a system may exhibit different types of response as the external forcing changes in amplitude and/or frequency. It also has been shown how an analytical study based on the multiple-scale analysis is able to describe this kind of behavior.

Thanks to these initial observations, it can be stated that the forcing term plays a crucial role in the problem. In this paper the analytical treatment is slighlty different than in [START_REF] Gourc | Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink[END_REF], in order to take into account the forcing term since the very first steps of the multiple-scale expansion. As for the experimental part, in order to prove the efficacy of the absorber for different frequencies, the same VI-NES as in [START_REF] Gourc | Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink[END_REF] is used, but applied to a different primary system having a higher natural frequency. Moreover, the use of a bearing slider is avoided as it turned out being a source of perturbation for the experiments. Four levels of force are tested and for each level a frequency sweep is performed; this leads to a study of the response of the VI-NES over a considerable range of forcing cases.

Finally, the analytical model is used to explain the behavior experimentally observed and a comparison between the experimental and the analytical results is presented. In addition, the influence of the design parameters of the VI-NES on the system's response is discussed thanks to some analytical examples.

Experimental tests

The experimental study has been conducted aiming to observe the behavior of the system and to explore the existing different types of response which can arise and how they are related to the external forcing in terms of magnitude and frequency. This phase of experimental investigation will be subsequently exploited and the observations explained by the analytical study of the system.

The experimental apparatus is shown in Fig. 1. It is constituted of a singledegree-of-freedom linear oscillator (LO) to which the VI-NES is attached. The LO is harmonically forced by an electrodynamic shaker. Figure 2 displays the tested prototype and its schematic: the primary mass M is connected to its base by means of two flexible blades. The base is then screwed to the vibrating table. On the primary mass, a cylinder is placed and the small VI-NES mass is free to roll inside. The cylinder is pierced on the sides in order to make the air effect negligible and to visually observe the motion of the ball. Firstly a dynamic identification of the primary system is performed. The system is excited by a swept-sine external force with constant amplitude and the displacement of the primary mass is measured by means of a Laser Doppler Vibrometer (LVD). The use of a LVD rather than an accelerometer has been dictated by the fact that the impacts would drastically perturb the signals delivered by an accelerometer. The modal parameters of the primary system are shown in Tab.1 and the mass values and their ratio in Tab.2. Several tests were performed on the primary structure at different levels of base acceleration between 0.1 -0.5g. They showed that the primary system was not perfectly linear and that a slight variation in the modal parameters occurred as the external force varied. However this was a weakly nonlinear behavior that did not affect the purpose of the tests. It is important to highlight the particularly small mass ratio between the VI-NES and the primary system: less than 1%. Once the primary system has been identified, we arm the VI-NES by inserting the small mass into the cavity and we carry out some swept-sine tests for four different levels of forcing: [0.2; 0.3; 0.4; 0.5]g; the displacement is measured by the LVD and the spectra are then obtained. The fact of using a large-mass shaker permits the shaker not

f 0 [Hz] K[N/m] λ[N s/m] ξ
M [kg] m[kg]
= m/M 3.807 0.032 0.84% Table 2: Mass values and their ratio . to be perturbed when it goes through the natural frequency of the LO during the swept-sine. Fig. 3 shows the displacement spectra for the system with and without the VI-NES for four levels of external excitation: [0.2; 0.3; 0.4; 0.5]g. Fig. 4 shows the measured displacement for two different frequencies at the same amplitude of external acceleration.

Firstly, we can observe that a threshold in the amplitude of the primary mass' oscillations exists for the VI-NES to become active: we will refer to it as activation threshold. Secondly, the system can exhibit two types of response and, depending on the magnitude and the frequency of the external forcing, either one or another may appear.

The responses can qualitatively be classified as:

• -Idle VI-NES: no impacts occur and the primary system's dynamics is not perturbed by the presence of the VI-NES: the amplitude of the LO's oscillations is lower than the activation threshold.

• -Constant Amplitude Response (CAR): the VI-NES is stably active and the amplitude of the primary mass displacement remains constant (fig. 4 left).

• -Strongly Modulated Response (SMR): the primary system goes through alternatively increasing and decreasing amplitude cycles and as a consequence the fast oscillations appear to be modulated. This behavior is caused by a cyclical activation/deactivation of the VI-NES (fig. 4 right). The VI-NES seems to well accomplish its task as a vibration absorber since the amplitude of the response is reduced nearby the resonance of the primary system. This is the proof that a Targeted Energy Transfer occurs from the LO towards the VI-NES and that the energy is then dissipated by the impacts. It is important to highlight that this goal has been achieved although a proper sizing process of the VI-NES has not been carried out and with a significantly small mass ratio = 0.84%. This result proves that the VI-NES is able to automatically tune itself to the primary system. This is a relevant general feature of nonlinear absorbers due to their lack of a natural frequency. However, whereas for a cubic stiffness NES the absence of a natural frequency is a delicate goal to reach [START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF], for a VI-NES it appears to be as an evident outcome. Still, we remind that the primary objective of this experimental study was not to seek the optimal performance but rather to investigate the qualitative behavior of the VI-NES.

f [Hz]
Looking at the spectra of Fig. 3 one can draw the conclusion that there exists a criterion on primary mass displacement in order to activate the VI-NES. Indeed a threshold in amplitude/energy is observed beyond which the VI-NES goes through a 1:1 resonance with respect to the LO. The terminology 1:1 resonance is used to maintain a correlation with the cubic stiffness NES, even though in this context it means that two impacts per oscillation cycle of the LO occur. This can be seen in Fig. 5 where the signal of an accelerometer placed on the primary mass is shown. The peaks represent the shocks occurring two times per period throughout the oscillations of the LO.

Fig. 6 shows a map illustrating the nature of the observed responses as a function of the magnitude and the frequency of the external forcing. For a low level of external forcing only a SMR is registered through the whole range of frequencies where the VI-NES is active. As the external force increases, a CAR appears at the lowest frequencies of the activation range and the SMR tends to disappear. When high levels of external forcing are reached the CAR becomes the prevalent, if not the only, type of response detected. It is therefore clear that for a given VI-NES the kind of response which might arise is a function of the magnitude and the frequency of the external forcing. In the next section this behavior will be explained by an analytical analysis of the mathematical problem.

Analytic treatment

In this section the mathematical problem associated to the VI-NES is studied. An analytical treatment is presented, in which the multiple-scale method is used in order to obtain the Slow Invariant Manifold (SIM) and the fixed points of the problem. The multiple-scale is a perturbation method used to construct uniformly valid approximations to the solution of the problem under consideration. This is done by introducing fast-scale and slow-scale variables and treating these variables as independent. This method is particularly useful when the problem is characterized by having multiple dynamics, each with their own scales. In our problem one scale is associated to the fast oscillations and another to their amplitude's slow modulation.

The approach followed is mainly based on the work of Gendelman [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF], where the analytic treatment of an unforced system with a VI-NES is presented. Although not original, these derivations are required for the sake of completeness of the exposition. Here we extend the treatment to a damped and forced system such as in [START_REF] Gourc | Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink[END_REF], with the difference that the forcing term is not hypothetically supposed to be of order 1 but considered since order 0 in the multiple-scale expansion.

The schematic of the model is presented in Fig. 2. We define the variables u, v, and x e as the displacements of the primary mass M , of the NES mass m and of the base respectively. Before going through the study of the equations of motion, we model the impacts by using the Newton's experimental law and the momentum conservation principle. This model, although quite simple, is able to represent the physics of the impacts by means of only one parameter: the restitution coefficient r.

u(t + j ) -v(t + j ) = -r( u(t - j ) -v(t - j )) (1)
M u(t + j ) + m v(t + j ) = M u(t - j ) + m v(t - j ) (2)
Where t + j and t - j are the time instants after and before the j th impact respectively. Eq.( 1) provides a relation for the relative velocities of the two colliding masses after and before the impact by use of the restitution coefficient 0 < r < 1. Then, in the limit cases, impacts can be characterized as perfectly elastic r = 1 or perfectly inelastic r = 0. Eq.( 2) expresses the momentum conservation throughout an impact. Each mass experiences an instantaneous momentum variation that can be represented by a Heaviside step function. The derivative of momentum with respect to time is the force acting through the impact. Thus in (3) we express the force each mass is subjected to as a Dirac delta function, which represents the distributional derivative of the Heaviside step function. In Eq.(3) the summation includes all the impacts that occur over the time. Since all the terms of the summation are Dirac delta functions, it means that they are always zero except when an impact occurs. When this happens, only the one term of the sum associated to that specific impact takes a value different than zero, as its delta becomes one.

Finally the equations of motion can be written as:

M ü + λ u + Ku + M m(1 + r) M + m j ẇ-δ - j = Kx e + λ ẋe mv - M m(1 + r) M + m j ẇ-δ - j = 0 (3)
Where M , λ and K are the mass, damping and stiffness of the primary system, m is the mass of the VI-NES and r the restitution coefficient.

Or:

ü + 2ω 0 ξ u + ω 2 0 u + m(1 + r) M + m j ẇ-δ - j = ω 2 0 x e + 2ωξ ẋe v - m(1 + r) M + m j ẇ-δ - j = 0 (4)
Where ω 2 0 = K/M , 2ω 0 ξ = λ/M and = m/M . Defining the barycentric coordinates X and w 1

X := u + v w := u -v (5)
And the following new variables:

τ := t ω 0 √ 1 + γ := 2ξ √ 1 + (6)
We end up with a system of two equations for X and w, where the subscript τ denotes the partial derivative ∂ ∂τ . They are up to now exact equations since no approximation has been made.

X τ τ + γX τ + γw τ + X + w = (1 + )x e + γ(1 + )x eτ w τ τ + γw τ + γX τ + w + X + (1 + r) j w - τ δ - j = (1 + )x e + (1 + )γx eτ (7)

Multiple-scale analysis

Assuming that the mass of the VI-NES m is small with respect to the primary mass M , = m/M << 1 can be used as a small parameter in the multiple scales analysis [START_REF] Nayfeh | Perturbation methods[END_REF].

τ k = k τ, k = 0, 1, . . . ; d dt = ∂ ∂τ 0 + ∂ ∂τ 1 + . . . ; X = X 0 (τ 0 , τ 1 , . . . ) + X 1 (τ 0 , τ 1 , . . . ) + . . . ; w = w 0 (τ 0 , τ 1 , . . . ) + w 1 (τ 0 , τ 1 , . . . ) + . . . ; (8)
1

The symbol := meaning equals by definition

Bearing in mind that << 1, the parameter γ can be expressed as a firstorder Taylor polynomial:

γ = 2ξ √ 1 + = 2ξ(1 + ) -1/2 2ξ(1 - 2 ) ( 9
)
The same is done with the external forcing term:

x e = X sin(ωt) = X sin ω ω 0 (1 + ) 1/2 τ x e X sin( ω ω 0 τ 0 ) + 1 2 ω ω 0 τ 1 X cos( ω ω 0 τ 1 ) (10)
Substituting ( 8), ( 9) and ( 10) into [START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF] and only keeping the zero-order terms:

∂ 2 X 0 ∂τ 2 0 + 2ξ ∂X 0 ∂τ 0 + X 0 = X e0 + 2ξ ∂X e0 ∂τ 0 ∂ 2 w 0 ∂τ 2 0 + 2ξ ∂X 0 ∂τ 0 + X 0 + (1 + r) j ∂w - 0 ∂τ 0 δ - j = X e0 + 2ξ ∂X e0 ∂τ 0 (11)
Where X e0 = X sin(Ωτ 0 ) with Ω = ω ω0 . By considering the experimental values we identified (ξ ≈ ≈ 0.8%) we can reasonably assume that ξ = λ 2ω0M has the same order as and therefore that ξ = O( ). Consequently, the terms multiplied by ξ in Eq.( 11) can be neglected. That yields:

∂ 2 X 0 ∂τ 2 0 + X 0 = X e0 ∂ 2 w 0 ∂τ 2 0 + X 0 + (1 + r) j ∂w - 0 ∂τ 0 δ - j = X e0 (12)
Alternatively, one can say that solutions of Eq.( 12) will be very close to solutions of Eq. [START_REF] Gendelman | Bifurcations of self-excitation regimes in a van der pol oscillator with a nonlinear energy sink[END_REF] given the small influence of ξ. Then the two equations of Sys.( 12) can be solved for X 0 and w 0 . Here we follow the ansatz given by Gendelman in [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF] and [START_REF] Gendelman | Dynamics of forced system with vibroimpact energy sink[END_REF] with the exception that the forcing term is present at the 0 order. That leads to the following expression for X 0 :

X 0 = C(X, Ω, τ 1 ) sin(Ωτ 0 + ψ(τ 1 )) (13)
C and ψ are the amplitude and the phase of X 0 . The amplitude C is a function of the external forcing X e0 and terms of order-1. The phase ψ is a function of order-1 terms as it depends on damping ξ. Substituting [START_REF] Lee | Suppressing aeroelastic instability using broadband passive targeted energy transfers, part 2: Experiments[END_REF] in the second equation of ( 12) we obtain:

∂ 2 w 0 ∂τ 2 0 + (1 + r) j ∂w - 0 ∂τ 0 δ - j = A sin(Ωτ 0 + θ) (14)
Where A sin(Ωτ 0 + θ) is the sum of X 0 and X e0 . These two terms are two sinus with the same frequency Ω and shifted in phase by ψ. It can be shown that the sum of them is a single sinus with a new amplitude A and phase θ to be determined.

A solution for w 0 in the following form is sought:

w 0 = - A(C, X, ψ) Ω 2 sin(Ωτ 0 + θ) + f (τ 0 , τ 1 ) (15)
Inserting ( 15) into ( 14):

∂ 2 f ∂τ 2 0 + (1 + k) j ∂f ∂τ 0 τ - 0 - A Ω cos(Ωτ 0 + θ) δ - j = 0 ( 16
)
Assuming that TET occurs in an established state of 1:1 resonance between the VI-NES and the LO and neglecting the transient process leading to 1:1 resonance, the particle will move symmetrically and at the same frequency of the LO. A form of f representing the "free flights" interrupted by the impacts is:

f (τ 0 , τ 1 ) = 2α π arcsin (cos(Ω(τ 0 -η))) (17)
Where t j = η + πj, j = 1, 2, . . . are the unknown impact instants. A schematic representation of f is presented in Fig. 7. Substituting ( 17) into ( 16) and integrating over a small interval around time τ 0 = η:

- 4αΩ π + (1 + r) 2αΩ π - A Ω cos(Ωη + θ) = 0 (18)
And finally:

-A cos(Ωη + θ) = Ω 2 2(1 -r) π(1 + r) α = Ω 2 σα (19)
Where σ = 2(1-r) π(1+r) . One more relation between A and α can be found by considering the expression (15) evaluated when an impact occurs: τ 0 = η and w = ±L.

- A Ω 2 sin(Ωη + θ) + α = L (20)
Finally we have:

A Ω cos(Ωη + θ) = σα A Ω sin(Ωη + θ) = L -α (21)
With A Ω = -A/Ω 2 . The two equations ( 21) allows one to analytically define the Slow Invariant Manifold of the problem:

α = L ± L 2 -(1 + σ 2 )(L 2 -A 2 Ω ) 1 + σ 2 Or: α = L ± √ 1 + σ 2 A 2 Ω -A 2 Ωmin 1 + σ 2 (22)
Where A Ωmin = σL √ 1+σ 2 is a minimum value of amplitude A/Ω 2 for a 1:1 resonance to be established. 17) (blue) and its derivative (red).

Eq.( 22) defines the Slow Invariant Manifold of the system and a graphical representation for the couple of parameters r = 0.65 and L = 15 mm is displayed in Fig. 8. A first important piece of information we can draw from Eq.( 22) is that a minimum for A Ω exists for solutions to appear. In order to have a physical meaning of the two variables of the SIM, A Ω and α can be thought as respectively the LO's and the NES' oscillation amplitudes. Then the lower threshold is actually a minimum amount of LO's energy (oscillations amplitude) the VI-NES needs to turn active. As previously mentioned this is a typical feature of nonlinear systems [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators part ii resonance capture[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators i: dynamics of the underlying hamiltonian system[END_REF][START_REF] Kerschen | Irreversible passive energy transfer in coupled oscillators with essential nonlinearity[END_REF]. Moreover, information about stability of solutions can be obtained from 0 -equations. Indeed a stability analysis can be performed by means of the Poincaré Map which reveals that the SIM is composed by two branches of solutions: one stable and one unstable. We will not describe the analysis here but only suggest that the reader refer to [START_REF] Li | Bifurcations and chaos in a forced zero-stiffness impact oscillator[END_REF] for a detailed mathematical description. Indeed, to have the same formal problem as in [START_REF] Li | Bifurcations and chaos in a forced zero-stiffness impact oscillator[END_REF] we just need to rearrange the second equation of Sys. [START_REF] Lee | Suppressing aeroelastic instability using broadband passive targeted energy transfers, part 1: Theory[END_REF] as:

∂ 2 w 0 ∂τ 2 0 = A sin(Ωτ 0 + θ) |w 0 | < L ∂w + 0 ∂τ 0 = -r ∂w - 0 ∂τ 0 |w 0 | = L (23)
As it will be explained more in detail in the next section, from a mathematical point of view this threshold represents a saddle-node bifurcation beyond which two branches of solutions appear, one stable and one unstable.

The SIM strongly depends on the design parameters length L and restitution coefficient r as graphs in Fig. 9 show. Particularly, it is interesting to highlight how in the case of r = the value for A Ωmin and the corresponding value of α is equal to half the tube length, i.e. the distance gap between the ball and the tube before impacting. This is physically explicable as r = 1 means no loss of energy during the impacts, the state with two impacts per cycle could therefore be maintained without the primary mass moving, i.e. with no input of energy into the system. Once the SIM obtained, we push further our analysis and study Sys. ( 7) at the 1 scale in order to study the evolution of the system on the SIM as the amplitude and the frequency of the forcing term vary.

Only keeping the 1 -order terms, the first equation of Sys.(7) becomes:

∂ 2 X 1 ∂τ 2 0 + X 1 = -2 ∂ 2 X 0 ∂τ 0 ∂τ 1 -γ ∂X 0 ∂τ 0 - ∂ 2 X 0 ∂τ 2 0 -w 0 + Y cos(Ωτ 0 + β(τ 1 )) (24)
Where Y cos(Ωτ 0 + β(τ 1 )) = X 0 cos(Ωτ 0 ). After substituting ( 13) into [START_REF] Gourc | Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink[END_REF] and eliminating secular we are able to find the approximate solution provided by the multiple-scale analysis. By equaling derivatives with respect to time to zero, the fixed points are obtained. These steps are not reported here for the sake of conciseness and can be found in [START_REF] Gourc | Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink[END_REF]. Finally we obtain a fourth order polynomial relating A and α:

a 2 A 4 + a 1 A 2 + a 0 = 0 ( 25
)
Where the coefficients a 0 , a 1 , a 2 are functions of α:

a 0 = 64 B 2 π 4 σ 2 + 1 α 4 - 128 B 2 π 4 α 3 + 64 B 2 π 4 α 2 (26)
a 1 = 32 B 2 π 2 µσ 2 + δ α 2 - 32δ B 2 π 2 α -1 a 2 = (µ 2 + 4δ ) B 2
Where δ is the detuning parameter defined as Ω = 1+ δ, B is the scaled external amplitude term B = X L and µ is the scaled damping coefficient µ = 2ξ/ . It can be seen that, unlike the SIM, the fixed points expression depends on the external excitation's amplitude X and frequency Ω through the variables B and δ. It is also important to notice that, as it was for the SIM, the fixed points expression is a function of the design parameters tube's length L and restitution coefficient r (through σ).

The solutions of the problem will satisfy both Eq.( 22) and Eq.( 25) and can graphically be identified by the intersections of these two curves on the plane A Ω -α. Alternatively, one can say that the fixed points of Eq.( 25) have to respect the constraints given by the SIM of Eq. [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF].

To summarize, through this section we mathematically described the physical problem, the impacts were modeled and the equations of motion obtained. The multiple scale method was applied to simplify and analytically solve the equations of motion. the assumption of 1:1 resonance between the LO and the VI-NES we found an expression for the Slow Invariant Manifold of the problem which provides some important information about the existence and the stability of all the possible solutions. Subsequently, by developing the equations of motion up to 1 -order, the fixed points were calculated, i.e. the steady-state solutions for a given external forcing. These points should respect the conditions imposed by the SIM to be actual physical solutions.

Experimental and analytical results comparison

The purpose of this section is to find an analytical description and explanation to the VI-NES' behavior we observed experimentally. Particular attention is paid to the type of response the VI-NES may exhibit depending on the magnitude and frequency of the external forcing. Three levels forcing are considered: low, medium high. For each level the evolution the system on SIM as the external forcing frequency varies is analytically analyzed and compared to the experimental observations. For the analytic calculations, a coefficient of restitution r = 0.65 is used. This is a typical value for an impact involving two metal bodies. Each case is presented as a graph on the A Ω -α plan where the curves ( 22) and ( 25) have been plotted. It is important to notice that eq.( 25) can have real or imaginary solutions. In the graphs that follow only the real solutions are plotted as they represent the only physical fixed points.

Low level forcing

For a low level of forcing (ẍ e = 0.1g) the activation threshold condition is not respected at any frequency and the VI-NES is never active. Experimentally the LO is not affected at all by the presence of the VI-NES. Analytically, as shown in fig. 10, the fixed points evaluated by solving Eq.( 25) (green curve) do not intersect the SIM curve and as a consequence no steady solution can be reached. The stable and unstable branches of the SIM are the blue and red-dashed lines respectively. The green curves are the solutions of [START_REF] Nayfeh | Perturbation methods[END_REF]. No intersections are present at any frequencies: the VI-NES is never active.

Medium level forcing

Fig. 11 shows the experimental spectrum the mass displacement for a base acceleration ẍe = The VI-NES stays idle until the primary oscillations reach the activation threshold, it is active for a range of frequencies nearby the natural frequency of the LO and it turns idle again for higher frequencies of the forcing. However, the only type of response observed is the SMR, suggesting the VI-NES is not able to establish a stable-state solution. This assumption is actually confirmed by looking at the evolution of the fixed points on the SIM in Fig. 12. Four different values of forcing frequency Ω = 0.97; 1.00; 1.01; 1.03 are shown. For Ω = 0.97 and Ω = 1.03 the isles (green curves) representing the fixed points do not intersect the SIM and the VI-NES is idle. For both Ω = 1.00 and Ω = 1.03 the fixed points curves have two intersection with the SIM. Nevertheless the points intersected are located on the unstable branch of the SIM: the solutions are then unstable. This is in agreement with the experimental observations which featured a Strongly Modulated Response. Analyzing the SMR, we can assume that the phase when the VI-NES is active is an unstable solution and therefore cannot be held. Subsequently the oscillations amplitude decreases until the VI-NES is not active anymore since the activation threshold condition is no more satisfied. Because of the constant external forcing the primary mass' oscillations get higher again and go beyond the activation threshold and the cycle starts over. Looking at Fig. 4, It should be noticed that the cycles are each time different and appear to be chaotic. Recently Gendelman et al. [START_REF] Gendelman | Dynamics of forced system with vibroimpact energy sink[END_REF] referred to this phenomenon as Chaotic Strongly Modulated Response (CSMR). A study of the chaotic SMRs is planned to be done in future work, where different chaos criteria might be applied to the experimental time series featuring SMR. The stable and unstable branches of the SIM are the solid-blue and red-dashed lines respectively. The green curves are the solutions of [START_REF] Nayfeh | Perturbation methods[END_REF]. A SMR regime is the only type of solution that can be reached. This is in agreement with the experimental results of Fig. 11. .

High level forcing

When the external acceleration is ẍe = 0.4g the spectrum showed in Fig. 13 presents two distinct zones featuring a CAR and a SMR. The VI-NES exhibits a CAR for most of the frequency range of activation and a SMR for a little portion just before turning back idle. The evolution of the fixed points on the SIM is illustrated in fig. 14. At Ω = 0.94 we can see that a second fixedpoints isle appears but that still there are no points of intersection with the SIM. At Ω = 0.97 this second isle intersects twice the SIM on the stable and on the unstable branch. It means that at least one stable solution exists and a CAR can be established. When the frequency goes up, the stable point tends to lose its stability and eventually becomes unstable (Ω = 1.04) as the isle of fixed points gets smaller. At Ω = 1.06 there are not intersections anymore and the VI-NES is idle again. In Sec.4.1 we saw how for a low level of forcing (ẍ e = 0.1g) the activation threshold condition is not respected at any frequency and the VI-NES is never active. A way to have solutions for this level of forcing would be reducing the tube's length. If this happens the SIM would move left on the A Ω -α plan and the fixed points could intersect it. This example is illustrated in Fig. 15 where the tube's length has been divided by two (L = L 0 /2 = 7.5mm).

The VI-NES is now capable to reach a Strongly Modulated Response state. If we go further in this analysis and keep reducing the length, there will be a point when the system gets to a Constant Amplitude Response. This example is shown in Fig. 16 where the length has been divided by three (L = L 0 /3 = 5mm)

We can assume that the smaller the tube's length is the easier it is to reach a stable solution (CAR), i.e. even with a low level of external excitation. However, it is highly important to point out that this is a discussion about the qualitative responses the VI-NES may exhibit and not about the efficiency of the VI-NES as an absorber. Here the problem is studied from a kinematic point of view, not involving energy aspects. It means that reducing the length and reaching a stable fixed point may potentially make the VI-NES less efficient as an absorber. This point can be better understood by studying the effect of a mass ratio variation. A similar analysis is conducted in the case where the mass ratio ε = m/M varies and the tube's length is kept constant. If the VI-NES mass is decreased to ε = 0.2%, a Strongly Modulated Response can be established, as shown in Fig. 17a. When the mass ratio is further decreased (ε = 0.15%) a Constant Amplitude Response state appears (see Fig. 17b). It should be clear that the effect of reducing the mass ratio allows the VI-NES to reach a stable state but at the same time reduces the level of energy dissipated by the shocks and consequently the VI-NES's efficiency as an absorber. For this reason and because of the complex dependency of the problem on these parameters, an extensive energetic study will be carried out in future investigations.

Conclusion

In this work the dynamical behavior of a harmonically forced 1-dof LO coupled to a VI-NES has been explored. An experimental study has been carried out, in which the different regimes of response that may arise (constant amplitude and strongly modulated) and their dependence on the external forcing have been analyzed. The experimental results have also shown that the VI-NES is able to significantly reduce the resonance peak of the primary system even in the case of a considerably small mass ratio ( = 0.84%) and with no need of an optimized designing process. The Targeted Energy Transfer mechanism is observed. Subsequently, the system has been analytically studied by means of the multiple-scale analysis. The Slow Invariant Manifold and the fixed points of the mathematical problem have been calculated. It has been shown that a threshold in the LO's amplitude for the TET to occur exist and how the SIM and the fixed points evolve as the external forcing term varies. The analytical model has allowed for a prediction of the response regime of the system and a good correlation with the experimental observations has been obtained.

These results complete the study begun in [START_REF] Gourc | Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink[END_REF] by providing an exhaustive experimental and analytical analysis of the VI-NES and its dynamics when applied to a 1-dof linear oscillator. This work will constitute the basis for further investigations on the VI-NES: an energetic study and an optimization of the performances as a vibration absorber will be the next steps to reach a complete characterization of this kind of device.

Figure 1 :

 1 Figure 1: The LO coupled to the VI-NES: the prototype installed on the vibrating table.

Figure 2 :

 2 Figure 2: The prototype (left) and its schematic (right).

Frequency

Figure 3 :

 3 Figure3: Experimental spectra of the primary mass displacement with (red) and without (blue) the VI-NES for four different levels of external forcing. It can be observed that the VI-NES is inactive for values of the primary mass' displacement under a certain threshold (≈ 2.8 mm). When the threshold is reached, the VI-NES turns active and two different regimes may occur: either a constant amplitude response (CAR) or a strongly modulated response (SMR).

Figure 4 :

 4 Figure 4: Time series: constant amplitude response -ẍe = 0.4g -f = 20.9Hz (left) and strongly modulated response -ẍe = 0.4g -f = 21.5Hz (right).

Figure 5 :

 5 Figure 5: Experimental impacts detection: acceleration of the LO measured by an accelerometer placed on the VI-NES in order to capture the impact instants. The peaks represent the shocks occurring throughout the oscillations of the LO, highlighted by the dashed line. A denotes the normalized acceleration of the LO. It can be seen as two impacts per cycle occur.

Figure 6 :

 6 Figure 6: Map showing the regime zones of the responses experimental observed as functions of the external excitation's magnitude and frequency.

Figure 7 :

 7 Figure 7: Sketch of function f according to eq.(17) (blue) and its derivative (red).

Figure 8 :

 8 Figure8: Slow Invariant Manifold -r = 0.65, L = 15mm. Two branches of solutions exist: one stable (solid blue) and one unstable (dashed red). A Ωmin is the minimum value of A Ω for the solutions to appear; this point mathematically represents a saddle-node bifurcation point.

Figure 9 :

 9 Figure 9: Evolution of 0 -SIM for constant L = 15mm as the restitution coefficient varies (left) and for constant r = 0.65 as the length varies (right).

Figure 10 :

 10 Figure10: Evolution of the fixed points for a low level of external forcing ẍe = 0.1g and four values of Ω. The stable and unstable branches of the SIM are the blue and red-dashed lines respectively. The green curves are the solutions of[START_REF] Nayfeh | Perturbation methods[END_REF]. No intersections are present at any frequencies: the VI-NES is never active.

Figure 11 :Figure 12 :

 1112 Figure 11: Experimental spectra of the LO's displacement with (red) and without (blue) the VI-NES. The regime of strongly modulated response (SMR) has been highlighted.

Figure 13 :Figure 14 :

 1314 Figure 13: Experimental spectra of the LO's displacement with (red) and without (blue) VI-NES for ẍe = 0.4g. The regimes of constant amplitude (CAR) and strongly modulated response (SMR) have been highlighted.

Figure 15 :Figure 16 :

 1516 Figure15: Same case as in Fig.10of low level forcing but for a tube's length divided by two (L = L 0 /2 = 7.5mm). It can be seen that now a SMR regime is possible.

Figure 17 :

 17 Figure 17: Same case as in Fig.10 of low level forcing but for a smaller mass ratio. It can be seen that a SMR and a CAR regime are possible.

Table 1 :

 1 Modal parameters of primary system.

	21.18	67421	8.566	0.008