Giovanni Tufano
email: giovanni.tufano@ec-lyon.fr

Fabrizio Errico

Olivier Robin

Christophe Droz

Mohamed Ichchou

Bert Pluymers

Wim Desmet

Noureddine Atalla

K-space analysis of complex large-scale meta-structures using the Inhomogeneous Wave Correlation method

Keywords: local resonances, wavenumber, dispersion curves, vibrations control, periodic structures, meta-structures

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In the aerospace sector, curved shells and axial-symmetric structures are widely used. The presence of stringers and frames in the longitudinal and circumferential direction of aircraft fuselages lets assume a certain degree of structural periodicity which, if properly designed, can give some advantages in the vibroacoustic response.

5

Composite materials are more and more used in this sector, to fulfill the requirements of reduced structural weight and high mechanical performances. The anisotropic behavior and the complexity related to their composite nature make the development of an analytical theory or of a predictive model very hard to obtain. The vibro-acoustic response and the dynamical behavior of laminated composite sandwich shells are of fundamentals relevance for the industry. The identification of the energy propagation, of the waves attenuation, with the related damping informations, and of the waves dispersions characteristics are relevant aspects to fully describe the vibro-acoustic behavior of a system [START_REF] Cherif | Damping loss factor estimation of two-dimensional orthotropic structures from a displacement field measurement[END_REF]. Some semi-analytical approaches have been developed in order to estimate the mechanical properties of sandwich composite structures [START_REF] Nilsson | Prediction and measurement of some dynamic properties of sandwich structures with honeycomb and foam cores[END_REF][START_REF] Chronopoulos | Modeling the response of composite panels by a dynamic stiffness approach[END_REF][START_REF] Chronopoulos | Efficient prediction of the response of layered shells by a dynamic stiffness approach[END_REF][START_REF] Droz | A new procedure for the determination of structural characteristics of sandwich plates in medium frequencies[END_REF].

These formulations are limited to the low-frequency domain, in which the modes are well defined; in the mid-high-frequency range, classical analytical approaches do not give a good estimation of the waves dispersion characteristics, due to an high modal density. As alternatives, in a wave propagation framework, for the parameter identification other methods based on the wavenumber domain (k -space) analysis [START_REF] Mcdaniel | Estimation of structural wave numbers from spatially sparse response measurements[END_REF][START_REF] Mcdaniel | A wave approach to estimating frequency-dependent damping under transient loading[END_REF][START_REF] Berthaut | K -space identification of apparent structural behavior[END_REF][START_REF] Ichchou | Multi-mode wave propagation in ribbed plates: Part I, wavenumber-space characteristics[END_REF] or based on the Statistical Energy Analysis (SEA) [START_REF] Lyon | Theory and Application of Statistical Energy Analysis[END_REF][START_REF] Langley | Some perspective on wave-mode duality in SEA[END_REF][START_REF] Chronopoulos | Predicting the broadband response of a layered cone-cylinder-cone shell[END_REF] are introduced. Some applications of the Prony's method to the wavenumber analysis of two-dimensional structures have been proposed [START_REF] Marple | Digital Spectral Analysis with Applications[END_REF]. The Prony technique is a means of identifying the constituent complex wavenumbers and their corresponding amplitudes in an exponential model of the spatial response. It is a two-steps process: the wavenumbers are first found and the amplitudes for these wavenumbers are then obtained from a linear least-squares fit of the model to the data. A modified extended technique based on Prony method to estimate the complex wavenumbers is shown in Grosh et al. [START_REF] Grosh | Complex wave-number decomposition of structural vibrations[END_REF]. Ferguson et al. [START_REF] Ferguson | The estimation of wavenumbers in two dimensional structures[END_REF] proposed a technique to identify the dominant wavenumber in a considered area, using a windowed field of the displacement of the structure, computing the correlation between the normal velocity measurement and an harmonic wave-field.

An algorithm for the identification of the complex wavenumber and of the damping loss factor was developed by Berthaut et al. [START_REF] Berthaut | K -space identification of apparent structural behavior[END_REF][START_REF] Ichchou | Multi-mode wave propagation in ribbed plates: Part I, wavenumber-space characteristics[END_REF][START_REF] Ichchou | Multi-mode wave propagation in ribbed plates. Part II: predictions and comparisons[END_REF]; the proposed method is called Inhomogeneous Wave Correlation (IWC) method and it allows to identify the complex flexural wavenumber in the broadband frequency range.

In order to increase the waves attenuation in certain frequency band or to generate some band gaps, periodic adds-on and locally resonant devices are widely used. The effect of wave propagation in ribs-stiffened plate structures is investigated in [START_REF] Ichchou | Multi-mode wave propagation in ribbed plates. Part II: predictions and comparisons[END_REF][START_REF] Orrenius | Calculation of wave propagation in rib-stiffened plate structures[END_REF], showing the multi-modal behavior in the wavenumber domain, in terms of uncoupled plate and stringer flexural wavenumbers. A vibroacoustic study of a fluid-loaded ribbed plate is shown in Maxit [START_REF] Maxit | Wavenumber space and physical space responses of a periodically ribbed plate to a point drive: A discrete approach[END_REF], where the response is investigated in the wavenumber domain using a discrete Fourier Transform based approach. A numerical investigation of the IWC method on curved shells and axialsymmetric structures is presented in [START_REF] Tufano | Wavenumber identification technique for axial-symmetric structures[END_REF], showing a good prediction of the angle-dependent wavenumber at different excitation frequencies and compared with the Fast Fourier Transform technique. For what concerns the stop bands behavior of tuned resonators, Claeys et al. [START_REF] Claeys | On the potential of tuned resonators to obtain low-frequency vibrational stop bands in periodic panels[END_REF][START_REF] Claeys | On the acoustic radiation efficiency of local resonance based stop band materials[END_REF] have demonstrated their potential use to reduce the vibrational response of panels, spatially distributing them over the panel surface. An application of the IWC method on a metamaterial plate with distributed resonators is given by Van Belle et al. [START_REF] Van Belle | On the impact of damping on the dispersion curves of a locally resonant metamaterial: Modeling and experimental validation[END_REF].

An experimental investigation on the vibroacoustical response of a cylindrical shell is presented in Williams et al. [START_REF] Williams | Experimental investigation of the wave propagation on a pointdriven, submerged capped cylinder using kspace analysis[END_REF], where the dominant wavenumbers are well identified in the regions close to the natural frequencies of the structure. An analysis in the wavenumber domain is used in Photiadis et al. [START_REF] Photiadis | Wave-number space response of a near periodically ribbed shell[END_REF] to investigate the acoustic response of a ribbed shell, demonstrating the dominant flexural behavior in the mid-frequency range.

Photiadis et al. [START_REF] Photiadis | The effect of internal oscillators on the acoustic response of a submerged shell[END_REF] investigated the effect of distributed internal oscillators on the acoustic response of a complex ribbed shell, showing their influence in the wavenumber domain compared to the bare configuration.

The effect of the internal frames on the vibroacoustic response of an aluminum cylinder is performed by Meyer et al. [START_REF] Meyer | Experimental investigation of the influence of internal frames on the vibroacoustic behavior of a stiffened cylindrical shell using wavenumber analysis[END_REF], providing a better understanding of frames' influence in the wavenumber domain and on the dispersion curves, mainly based on the numbers and the spatial distribution of the internal frames.

Nateghi et al. [START_REF] Nateghi | Wave propagation in locally resonant cylindrically curved metamaterial panels[END_REF][START_REF] Nateghi | Design and experimental validation of a metamaterial solution for improved noise and vibration behavior of pipes[END_REF] have discussed and experimentally validated that by adding locally resonating structures to cylindrically curved panels and pipes, improved noise and vibration behavior can be obtained in a targeted and tunable frequency band. This concept was applied on an aircraft sidewall panel by Droz et al. [START_REF] Droz | Improving sound transmission loss at ring frequency of a curved panel using tunable 3D-printed small-scale resonators[END_REF] , improving the acoustic properties of the panel in the ring frequency region, showing how a smart design of metamaterial structures can be adopted to improve the vibroacoustic performance.

This work is fully devoted to the experimental identification of the dispersion characteristics of these complex structures, subjected to either a diffuse acoustic field or a point mechanical excitations. Each considered structure is first tested alone, and then with the addition of spatially-distributed small-scale resonators. The method used in this work is an extended version of the IWC method, introducing the effect of the local dynamics of the resonators and the effect of the acoustic excitation, in presence of periodic adds-on. The main advantages of this method are the possibility to investigate both in-plane and curved structures, the identification of the resonators related band gaps and the use of the experimental measured data, without any restriction related to an ordered mesh-grid for the acquisition points.

In section 2, an overview of the IWC method is presented and an extended formulation to take in account the curvature effect is developed. The experimental set-up for each studied case is fully described in section 3, showing how the experimental tests were performed and giving information about the tuning frequencies of the resonators. The section 4 is fully devoted to the results discussion; the isotropic steel panel is first considered, then all the results concerning the curved structures are discussed. A numerical simulation on an isotropic curved panel is performed to validate the feasibility of the proposed IWC formulation for curved structures, followed by experimental results on the two considered curved structures for tests.

Overview of the Inhomogeneous wave correlation method and new formulation for curvature effect

The method developed in this work is derived from the Inhomogeneous Wave Correlation (IWC) method.

The inverse approach here proposed is extended to complex curved structures in presence of periodic elements and small-scale resonators. The aim of this approach is the complete description of the angle-dependent dispersion characteristics and the identification of the structural damping information of a vibrating structure, using the whole wave field of a vibrating structure as primary input [START_REF] Berthaut | K -space identification of apparent structural behavior[END_REF][START_REF] Ichchou | Multi-mode wave propagation in ribbed plates: Part I, wavenumber-space characteristics[END_REF][START_REF] Ichchou | Multi-mode wave propagation in ribbed plates. Part II: predictions and comparisons[END_REF][START_REF] Berthaut | Contribution à l'identification large bande des structures anisotropes: application aux tables d'harmonie des pianos[END_REF][START_REF] Inquiété | Numerical simulation of wave propagation in laminated composite plates[END_REF]. This method calculates the correlation between the energy carried on by an inhomogeneous traveling wave and the total energy of the vibrational field. Considering a plane structure and a plane wave approach, the inhomogeneous wave can be expressed as follows: where θ is the angle of propagation and k C = k + ik is the complex wavenumber, which contains both propagation and attenuation informations. At each frequency f and for a given direction θ, the unknown complex wavenumber k C can be identified as the location of the maximum of the normalized correlation function, denoting the point where the measured signal ŵ correlates best with the inhomogeneous wave wk C ,θ (x, y):

wk C ,θ (x, y) = exp{-ik C (θ)(x cos θ + y sin θ)} (1)
I(k C , θ) = Ω ŵ • w * k C ,θ dx dy Ω | ŵ| 2 dx dy • Ω | wk C ,θ | 2 dx dy (2)
where * denotes the complex conjugate. Rewriting Eq. 2 in the discrete domain, the integration over the entire surface Ω is replaced by a finite weighted sum:

Ω dx dy =⇒ N j=1 h j Ω j (3)
where h j is the coherence of the measured signal at each point (h j = 1 if the coherence is not available), Ω j is an estimation of the surface around the point j and N is the total number of acquisition points. Introducing

Eq. ( 3) into Eq. ( 2), the correlation function becomes

I(k C , θ) = N j=1 ŵ(x j , y j ) • w * k C ,θ (x j , y j )h j Ω j N j=1 | ŵ(x j , y j )| 2 h j Ω j • N j=1 | wk C ,θ (x j , y j )| 2 h j Ω j (4)
The algorithm first puts angle θ into a discrete set of values; for each of these angles, the maximum of IWC is located at a value (k , k ), creating two functions θ → k (θ) and θ → k (θ) defined on a set of discrete values of θ. A schematic representation of the IWC algorithm is shown in Fig. 1. From the identified structural wavenumber, k, the k x and k y components in the x and y direction are defined by

k x = k cos θ k y = k sin θ (5)
The full complex dispersion relation is calculated by a repetition of the algorithm on the whole frequency band of interest.

For point harmonic excitations, the estimate of the decaying part of the wavenumber can be improved by taking in account the location of the excitation (x 0 , y 0 ) [START_REF] Van Belle | On the impact of damping on the dispersion curves of a locally resonant metamaterial: Modeling and experimental validation[END_REF]. The spatially decaying plane wave assumes the following expression:

wk C ,θ (x, y) = exp{-ik C (θ)((x -x 0 ) cos θ + (y -y 0 ) sin θ)} (6)
Based on this initial description, a new formulation for curved structures can be derived, which takes in account the curvature effect in the IWC formulation. From the notations and the coordinates system shown in Fig. 2, identifying with R the curvature radius of the structure and with ϕ the angular coordinate, the following relationships can be derived:

y = R sin ϕ ≈ Rϕ and k ϕ = Rk y (7)
where a small value of the angle ϕ is assumed (this value strictly depends on the discretization resolution).

Substituting these relationships in the expression of the inhomogeneous wave, we obtain:

wk C ,θ (x, ϕ) = exp{-ik C (θ)((x -x 0 ) cos θ + (ϕ -ϕ 0 ) sin θ)} (8)
where ϕ 0 is the angular position of the excitation point. Moving from the Cartesian coordinate system to the curved one, Eq. 4 is modified as follows

I(k C , θ) = N j=1 ŵ(x j , ϕ j ) • w * k C ,θ (x j , ϕ j )h j Ω j N j=1 | ŵ(x j , ϕ j )| 2 h j Ω j • N j=1 | wk C ,θ (x j , ϕ j )| 2 h j Ω j (9)
The location of the maximum of the correlation function I(k C , θ) gives the identified complex wavenumber in the specified direction. When the structure is complex, the correlation function could be characterized by an absolute maximum and some other local maxima, denoting the presence of different bending waves propagating at the same time. Following the evolution of the local maxima, the multi-modal behavior of the structure can be described, as shown in section 4.3.

Experimental methods

Three different structures have been experimentally investigated: an isotropic steel plane panel, a thick curved composite laminated sandwich panel and a thin aluminum aircraft sidewall panel, including frames and stringers. The resonator is schematically represented in Fig. 3; the shape is similar to a cantilever beam, with a rectangular prism base of dimensions 10 x 10 x 8 mm 3 and a magnet glued on the opposite side, with the possibility to tune it by adding or removing the magnet masses. The resonators are 3D-printed using polycarbonate polymer, with a total dimensions of 33 x 10 x 8 mm 3 and a unit weight of 2.72 g. The resonator main beam has dimensions 23 x 10 x 4 mm 3 and the stiffener element has section of 4 x 3.5 mm. The concept of the resonators and their feasibility in reducing the bending wavenumber was demonstrated in [START_REF] Droz | Diffusion-based design of locally resonant sub-systems using a reduced wave finite element framework[END_REF].

The same concept was used in [START_REF] Droz | Improving sound transmission loss at ring frequency of a curved panel using tunable 3D-printed small-scale resonators[END_REF], in which they were also tested on the aluminum aircraft sidewall panel considered in the present work, with the aim to improve the sound insulation properties of the considered structure. A schematic representation of the resonator and its application on the host structure are shown in Fig. 3. A uniform distribution of the resonators on the surface of the structures is chosen to avoid any local and/or directional effect.

Isotropic plate

An isotropic steel plate of dimensions 0.95 m x 0.6 m and thickness of 2 mm is considered. The host structure has a total weight of 9 kg, approximatively. To simulate free boundary conditions, the structure was suspended at three points along the longer edge using Polylactic Acid filaments. The plate was excited by a point mechanical force using a Brüel & Kjaer shaker attached to the host structure using an impedance head (Fig. 4). The out of plane vibrational field was acquired using a Polytec PSV-500 Scanning Vibrometer.

An acquisition window of 0.92 m x 0.56 m was scanned, using a mesh of 60 x 24 along length and width, respectively, with a total number of 1440 acquisition points. The bare structure was first tested, and then the effect of distributed resonators tuned at frequencies of 800 Hz and 1400 Hz were successively tested. The resonators frequency being adjusted with varying tuning masses, the added weight was different for the two considered frequencies. For the first one (800 Hz), the added weight from the resonators represented 2.5% of the total panel weight, while for the second one (1400 Hz), the panel weight was increased by 1.5%.

The shaker and the resonators were mounted on the rear surface of the host structure while the front one was scanned by the Laser Doppler Velocimeter system. The excitation was generated using a white noise signal from 100 Hz to 2000 Hz. A schematic representation of the distributed resonators and of the experimental set-up is shown in Fig. 4.

Curved composite sandwich panel

The composite panel dimensions are 1.54 m x 1.62 m, with a curvature radius approximatively of 0.97 m, and thickness equal to 27.5 mm (Fig. 5a).

A preliminary experimental investigation of the dispersion characteristics of the composite laminated sandwich panel was first conducted. The panel has been simply hanged in the anechoic room assuming free boundary conditions. The tests are performed using an impact hammer and four accelerometers, placed in the circumferential and axial directions; the experimental data have been recorded using a Brüel & Kjaer real-time, multichannel sound and vibration data acquisition card. The experimental wavenumbers were calculated using the phase difference between the accelerometers, placed along the specific direction [START_REF] Rindel | Dispersion and absorption of structure-born sound in acoustically thick plates[END_REF][START_REF] Thwaites | Non-destructive testing of honeycomb sandwich structures using elastic waves[END_REF].

The recorded time data were filtered in the time domain to eliminate the reverberant field contribution.

The panel was then installed in the aperture between the coupled reverberant-anechoic rooms of the GAUS (Groupe d'Acoustique de l'Université de Sherbrooke) laboratory, at Université de Sherbrooke (Fig. 5).

The representative scheme of the coupled rooms is shown in Fig. 6. The reverberant room has dimensions The acoustic excitation was generated by a loudspeaker placed close to a corner of the room, with a white noise signal input from 100 Hz to 3000 Hz. Concerning the mechanical excitation, a Modal Shop Model 2025E shaker was installed in the reverberant room and attached to the test structure using an impedance head, as shown in Fig. 5b; the excitation was generated using a white noise signal from 100 Hz to 3000 Hz. The full vibrational field was measured using a Polytec 3D Scanning Vibrometer (PSV-3D), installed in the anechoic room, with dimensions 6.0 x 7.0 x 3.0 m 3 . The acquisition window was approximatively of 1.2 m x 1.0 m, in order to be sufficiently far from the boundaries and to reduce the influence of the boundary conditions, with a total number of 3111 measured points (mesh grid of 61 x 51).

A distributed pattern of 3D-printed small-scale resonators was then attached to the panel and compared to the bare test case (Fig. 5b). The chosen tuning frequency was 1000 Hz; the panel weight was increased by 4%

for a total number of 246 resonators. For each considered excitation (diffuse acoustic field (DAF) and point mechanical excitation), the bare structure was first tested, and then the effect of distributed resonators tuned at a frequency of 1000 Hz was tested. In the reverberant room, a control microphone was placed close to the surface of the excited structure to double check the incident pressure and normalize the FRFs (Frequency Response Functions).

Aluminum aircraft sidewall panel

The aluminum panel has dimensions 1.45 m x 1.70 m, with a curvature radius approximatively of 1.30 m and a thickness of 1.2 mm (Fig. 7). The panel was installed in the aperture between the coupled reverberantanechoic rooms of the GAUS laboratory, already described in the section 3.2. For the point mechanical excitation, the bare panel was first tested with free boundary conditions. For the diffuse acoustic field excitation, the bare structure was first tested and the effect of distributed resonators tuned at three different The presence of the 246 resonators increased the panels weight by 4.5% in this case.

The same Modal Shop Model 2025E shaker, described in the section 3.2, was used to excite the aluminum panel, using a white noise signal input from 100 Hz to 2000 Hz. The structure was freely suspended to simulate free boundary condition; the shaker was placed on an internal frame structure, as shown in Fig. 8a. The external surface was scanned using a Polytec 3D Scanning Vibrometer, with an acquisition window of approximatively of 0.85 m x 0.95 m, with 5451 acquisition points (equivalent to a mesh grid of 69 x 79).

To test the aluminum panel under diffuse acoustic field excitation, the panel was installed between the two rooms, as shown in Fig. 7. The resonators were attached on the external surface (in the reverberant room) and the internal surface was scanned by a Polytec 3D Scanning Vibrometer, in the skin regions between stringers and frames, as shown in Fig. 9a; the total number of scanned points was 2916, with a mesh grid of 27 x 9 for each region (total mesh grid of 54 x 54).

Numerical validations and experimental results

Firstly, the proposed IWC formulation for curved structures is numerically validated. The method is applied on an isotropic curved panel; the employed material is a standard aeronautical aluminum alloy, is validated by the Wave Finite Element Method (WFEM) [START_REF] Brillouin | Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices[END_REF][START_REF] Mace | Finite element prediction of wave motion in structural waveguides[END_REF][START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF][START_REF] Errico | The modelling of the flow-induced vibrations of periodic flat and axial-symmetric structures with a wave-based method[END_REF] ; the dispersion curves are in agreement on the whole frequency range, as shown in Fig. 10b.

Steel plate: experimental results and validation

In this case, the dispersion curves obtained with the IWC method is compared with the classical Kirchhoff's thin plate theory. An analytical expression of the dispersion relation can be derived from this theory, obtaining the following expression for the flexural wavenumber

k = √ ω ρh D 1/4 ( 10
)
where ρ is the mass density, h is the plate thickness and D is the bending stiffness.

The dispersion curves are estimated for all tested configurations and compared with the analytical solution, as shown in Fig. 11. The estimated dispersion curves are in agreement with the analytical solution in the whole frequency range. In Fig. 11a, the effect of the resonators tuned at 800 Hz is shown, giving a band gap of approximatively 200 Hz width; before and after the band gap, the dispersion curve follows the one of the bare configuration and the analytical solution. For what concerns the resonators tuned at 1400 Hz, the originated band gap width is of approximatively 250 Hz, as shown in Fig. 11b. For both configuration, a strong attenuation is observed in the resonators tuning frequency regions.

Curved composite laminated sandwich panel: experimental results and validation

Firstly, the Frequency Response Functions (FRFs) of the two configurations of the panel are compared, for both excitations (Fig. 12). For what concerns the FRF of the panel under diffuse acoustic field excitation, the natural modes are not well identified due to the high damping level; in any case, around the tuning frequency of the resonators a drop of amplitude is observed (Fig. 12a). Under mechanical excitation, the natural modes of the structure can be easily identified; the drop around the tuning frequency of the resonators is clearer, as shown in Fig. 12b.

The measured displacement fields for all four configurations and for both excitation condition are plotted in Fig. 13, choosing a frequency around the tuning frequency of the resonators. To make a comparison between the experimental displacement fields, all the displacements have been normalized by the maximum value.

Fig. 14 shows the bending dispersion curves under diffuse acoustic field excitation for both configurations, with and without resonators; the dispersion curves are computed in the two orthogonal directions, axial and circumferential. For both directions, the local dynamics of the resonators is well described.

The same comparison is done using the mechanical excitation, as shown in Fig. 15; the dispersion curves are calculated in both directions, with and without the resonators. The local dynamics of the resonators is well described in both directions.

From the estimation of the complex wavenumbers, a calculation of the damping loss factor is performed, using the following relationship between the real and imaginary part of the wavenumber

η = k 4 k 4 (11)
The experimental damping loss factor is calculated for both configuration, the test case and the panel with distributed resonators, as shown in Fig. 16a, where the damping loss factor is expressed in one twelfth octave band. In low frequency region, the damping loss factor appears very high, mainly due to the boundary conditions influence. Around the tuning frequency of the resonators an increase of the damping factor is present; this effect is correlated to the continuously increasing of the imaginary part of the wavenumber and decreasing of the real one, consequently to a strong reduction of the vibration level.

A different method derived from the Statistical Energy Analysis (SEA) is used to estimate the damping loss factor: the power input method (PIM). The basic idea is that, in steady-state condition, the input power is equal to the dissipated power [START_REF] Cherif | Damping loss factor estimation of two-dimensional orthotropic structures from a displacement field measurement[END_REF]. The damping loss factor is expressed by

η f = P f ωE f (12)
where P f is the input power and the E f is the total energy, averaged in the frequency band. Input power and total energy are calculated by the following expressions, respectively

P f = 0.5 • (F * v) = 0.5 • ω2 ω1 G F v (ω) dω (13)
E f = M f v 2 f = M f ω2 ω1 G vv (ω) dω (14)
where G F v is the cross-spectral density between the force F and the velocity v, M f is the mass related to the measurement area and G vv is the auto-spectral density of the velocity. The quadratic velocities and the force are averaged over 200 points. All these quantities are derived from the use of an impedance head.

Aluminum aircraft sidewall panel: experimental results and validation

The finite element model of the panel is built and shown in Fig. 17; the numerical displacement field is obtained by a full FE analysis, using a point harmonic excitation at the same location of the shaker in the experimental test. The material properties introduced in the numerical model are of a standard aluminum alloy for aeronautical application: Young's modulus E = 73.1 GPa, mass density ρ = 2780.0 kg m -3 and Poisson coefficient ν = 0.33. The structure is not modeled with a very high degree of fidelity. For example, the all the joints and connections are not modeled, stringers and frame are considered rigidly connected; the small air-gap present in the real structure is neglected and some small holes in the frames structure are not present in the numerical model, because their effect on the dispersion characteristics is considered of small influence. The FE model is built using shell elements (ANSYS SHELL181); the total number of structural elements is approximatively of 3.2 • 10 4 . The normal displacements of the nodes of the external surface are used as primary input in Eq. 9; from the full-field measurement, the multi-modal behavior is well identified in the numerical simulation and experimental test, with shaker excitation and full-field acquisition, as shown in Fig. 18a. The two different modes of propagation are quite evident in the axial direction and they are well captured when the full-structure is scanned. The same behavior is not captured when the acquisition window is reduced to the internal bays (as shown in Fig. 9a), because the stringers vibration is not acquired.

Two uncoupled wavenumber are identified at the same frequency: plate flexural and stringer flexural. Some discrepancy between the numerical and experimental dispersion curves are present in Fig. 18a, mainly due to the modeling choices. A good agreement between the numerical dispersion curve and the experimental ones, under the two different loading conditions, is shown in Fig. 18, both in the axial and circumferential directions.

The effect of the resonators on the dispersion curves is remarkable, in both directions. In the axial direction (19a) the bending dispersion curve appears higher in frequency; in the circumferential one a band-gap between 500 Hz and 1000 Hz appears, as shown in Fig. 19b. In this frequency band, the energy flowing in structure is dissipated by the vibration of the resonators, showing an high apparent stiffening behavior (lower values of the wavenumbers compared to the bare test case).

Conclusions

In this paper, the feasibility of the Inhomogeneous Wave Correlation method on complex structures, in presence of a local dynamics effect and under diffuse acoustic field excitation have been demonstrated, giving a good estimation of the dispersion curves for all tested configurations. The global bending behavior of the aeronautical aluminum panel is well captured, even if the acquisition window is reduced to the bays between the stringers and the frames. At the same time, the local dynamics and the multi-modal behavior is not described. This is one of the main advantages of the proposed approach, which allows to describe the global behavior from a reduced acquisition window and a reduced number of points. The efficiency of this 3D-printed small-scale resonators is also demonstrated, obtaining a reduction of the vibrational field of the panels. For all tested configuration, in the frequency band the resonators are tuned to work in, a drop in the dispersion curves is observed. The method described in this work is mainly an energetic approach, which makes a correlation between the energy carried on by the inhomogeneous wave and the total energy of the vibrating structure. The drops in the dispersion curves appear because the energy flow in the structure is strongly reduced and dissipated by the resonators.

This preliminary study on the industrial application of small-scale resonators on real aeronautic structures opens the possibility to attenuate the vibration level in some specific frequency bands, working on the design

Figure 1 :

 1 Figure 1: Schematic representation of the IWC algorithm: vibrational field and propagative plane wave (a) and IWC correlation function (b)

Figure 2 :

 2 Figure 2: Curved structure coordinates system

Figure 3 :

 3 Figure 3: Schematic representation of the resonator (a) and its application on the host structure (b)

Figure 4 :

 4 Figure 4: Set-up of the steel plate with distributed resonators

Figure 5 :

 5 Figure 5: Curved composite laminate sandwich panel: bare configuration (a) and distributed resonators configuration (b)

Figure 6 :

 6 Figure 6: Illustration of the coupled rooms

Figure 7 :Figure 8 :Figure 9 :Figure 10 :

 78910 Figure 7: Set-up and installation of the bare configuration: outer skin subjected to DAF excitation (a) and inner surface (b)

Figure 11 :

 11 Figure 11: Dispersion curves comparison for the steel plate, with and without the resonators: at 800 Hz (a) and at 1400 Hz (b)

Figure 12 :

 12 Figure 12: FRFs comparison for both configurations and excitations: DAF excitation (a) and point mechanical excitation (b)

Figure 13 :Figure 14 :Figure 15 :Figure 16 :

 13141516 Figure 13: Normalized displacement fields, with and without the effect of the resonators: bare test case under DAF excitation (a), resonators configuration under DAF excitation (b), bare test case under shaker excitation (c) and resonators configuration under shaker excitation (d)

Acknowledgment

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 675441. The authors would like to acknowledge Maxime Bilodeau for his help in the installation and set-up of the 3D laser vibrometer. Patrick Levesque is also acknowledged for the installation of the panels in the reverberant room.


Jean-Marc Salotti
email: jean-marc.salotti@ensc.fr

Parametric

Parametric equation for the settlement of mars

Keywords: Mars settlement, colonization, self-sufficiency, parametric equation

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Is the settlement of Mars feasible or not? In general, how to determine if the settlement in space or on another planet is feasible and at what conditions? Many authors try to address that problem and provide partial answers [START_REF] Johnson | Space Settlements -A Design Study[END_REF][START_REF] Sagan | Pale Blue Dot: A Vision of the Human Future in Space[END_REF][START_REF] O'neill | The Colonization of Space[END_REF][START_REF] Hein | World Ships, Architectures & Feasibility Revisited[END_REF][START_REF] Zubrin | The Case for Mars: The Plan to Settle the Red Planet and Why We Must[END_REF][START_REF] Smith | An Adaptive Paradigm for Human Space Settlement[END_REF]. O'Neel addressed the problem of building a colony in space [START_REF] O'neill | The Colonization of Space[END_REF]. The total mass of what is to be sent to Earth orbit is so high that the feasibility seems very weak. For Mars, Zubrin explained that the presence of useful resources is a tremendous advantage [START_REF] Zubrin | The Case for Mars: The Plan to Settle the Red Planet and Why We Must[END_REF]. Water can be extracted from the ice, carbon from atmospheric carbon dioxide, iron from hematite, silicon from sand, etc. The Mars Homestead project illustrates this strategy [START_REF] Petrov | A Permanent Settlement on Mars: The Architecture of the Mars Homestead Project[END_REF] Nevertheless, there is a long way to go to be able to transform these resources into consumables and complex objects such as cars, computers and robots. As a factory cannot be run with very few people, a successful settlement would obviously require numerous persons and a long term financial investment. But how many persons are needed to achieve full autonomy or close to full autonomy on the planet? And if the number of persons is very high, e.g. several thousand people, how many tonnes of payload would have to be sent from Earth to sustain the lives of the settlers during the transition period? The feasibility is clearly linked to that mass, especially if no significant improvements are made neither for interplanetary propulsion systems nor for the costs of building and launching rockets from earth [START_REF]Global Human Mars System Missions Exploration, Goals, Requirements and Technologies[END_REF][START_REF] Salotti | Robust, affordable, semi-direct Mars mission[END_REF]. We propose to assume these two constraints and to address the difficult question of the feasibility. In Section 2, a simplified parametric equation is presented. In section 3, interpretations of several colonization scenarios are proposed in order to provide estimations for the parameters of the equation. The relevance of the model is discussed in conclusion and several ideas are suggested to improve it.

Looking for a model 2.1 Methodology

Let us assume that a human organization is committed to the project of settlement on another planet. In order to reach that goal, two important issues have to be addressed: How long will it take to succeed (autonomy achievement) and what would be the cost per year? In first approximation, the cost depends on two parameters: 

The cost per kilogram of material or object to be sent to the planet 

The total mass that has to be sent to the planet per year. We do not address here the cost per kilogram problem, we rather propose to focus on the total mass, which typically depends on two main parameters. The first one is simple, it is the number of astronauts. The second is hard to estimate, it is the mass of all needs except those that can be produced or created using in situ resources. Obviously, the total mass of needs grows with the number of astronauts, but the mass of objects or consumables produced on the planet also grows with the number of astronauts. Intuitively, as more and more tools and facilities can be shared when the number of astronauts increases, it is expected that the needs are quickly growing at the beginning of the settlement and are slowly decreasing when the number of astronauts becomes much higher. The problem is to be able to

IAC-19-A5,2,7,x49674
Page 2 of 5 determine the exact shape of that evolution curve. We propose to address the problem in a mathematical way.

Let n be the number of astronauts or settlers on the planet. The problem is to determine a function of variable n, called p(n), that computes the minimum payload mass that is required each year to sustain the lives of all persons. It is proposed here a two steps approach. First, a parametric model of that function is looked for and second, possible values for all parameters are suggested and discussed.

Proposed model

As a planet usually has numerous useful resources for humans, it is reasonable to assume that the minimum payload is achieved by sending only objects that cannot easily be produced on the planet using in situ resources. An important difficulty is to express what cannot easily be produced into mathematical terms. The ease to produce something generally depends on available tools, available energy and available human resources. However, without time constraints, energy production, human resources and the construction of new tools would not be a problem. Time can therefore be considered a key parameter of the model. In first approximation, the local production simply depends on the available working time of settlers. Nevertheless, as there are numerous acceptable living conditions and many different ways to produce objects and consumables, the optimal strategy is hard to find. One way or the other, a mathematical model should include parameters that would take this strategy into account. The problem can be formulated as follows (see equ. 1):

𝑝(𝑛) = 𝑘 * 𝑛 * ( 𝑟 𝑠(𝑛) -𝑤) (1)
Where:  w (in hours) is the individual working time capacity per year in average. This parameter may vary according to the type of work, the organization of the society, habits, etc. In modern societies, a person works approximately 2000 hours per year.  r (in hours) is the minimum individual working time requirement per year and in average to produce on the planet all objects and consumables that are necessary to sustain the life of one person. It includes agriculture time to grow plants, industrial time to extract chemical elements from the atmosphere and ores from the soil in order to produce metals, plastics and then tools and complex objects, as well as medical time, teaching time, administration time, etc. As all objects have a limited lifetime, the working time also includes the time to create or build each object divided by the lifetime of the object. The working time therefore includes a percentage of the construction time of buildings, cars, and all complex objects that are assumed to be required for living in a decent way.  s(n) is a function of n and is called the sharing factor.

For instance, if for 4 astronauts, on average, each object is shared between 2 persons, s(4) = 2. It is expected that the number of shared objects will grow with the number of persons. For instance, a kitchen may be shared by four or five persons of the same family and an electric power plant may be shared by a thousand people. Importantly, for a high enough value of n called nt, r/s(nt) equals w, which means that a threshold is reached, there will be enough people in the colony to produce everything. In other terms, the colony would potentially achieve full autonomy as soon as the number of persons exceeds nt.  Given w, r and s(n), the right part of the expression computes the time that is missing per person and per year to build all required objects to sustain the life of one person. It is multiplied by n to obtain the total missing time and by a coefficient called k to transform the missing time in tonnes.  k is a mass coefficient. If the missing time is greater than zero, it is necessary to compensate by sending objects from the Earth. A difficulty is to convert a time in mass. This is the role of constant k. It is obviously not required to send all missing objects. It is possible to send only tools or to provide parts of the missing objects, with the objective of minimizing the total payload mass and therefore k. A method is proposed in section 3 to determine k.

Scenarios

In this preliminary study, we did not try to determine the exact values of the parameters. We propose a simplified analysis with numerous assumptions and simplifications, which differ depending on the chosen scenario.

Scenario 1: Modern society

In a modern society, there are numerous complex objects such as computers, robots and other high tech devices for agriculture, metallurgy, transportation, etc. In order to produce these objects, it is necessary to extract iron, alumina, silicon, copper, etc. from different ores, to master plastics production, to build metallurgy factories, microelectronics factories, and finally to produce and assemble complex objects such as computers, cars, robots, etc. in other dedicated factories. In this scenario, as many people have to work in the industrial and digital worlds, it is expected that r is very high and k very small. The following parameters are proposed: -w=2000 hours r = 10 6 hours It means that 1 million hours of work would be required to produce all objects IAC-19-A5,2,7,x49674 Page 3 of 5 and consumables needed for a single person per year. s(n) = n  This is a simple way to consider the sharing factor. This function is interesting because it starts as expected with n equal to 1 whatever the value of  and it regularly increases with n but in less proportion, which is also expected. Intuitively, as for 2 persons more than 50% of all objects would be shared (house, car, life support system, facilities, etc.), it is assumed that is higher than 0.5. For the sake of simplicity, we propose 3 values for : 0.52, 0.54 and 0.6.

k = 0.00001 It is proposed to determine k by looking at p(1), which is the expected payload for a single person. For n=1, the sharing factor is equal to 1, which simplifies the equation. A single person would not be able to produce numerous objects or consumables on the planet. As most objects would have to be sent from Earth, an estimate of the payload can be made by looking at the payload mass that is expected for the first manned missions to Mars, which is in the order of 30 tonnes for 3 astronauts and a stay of 1.5 year on the planet. For the sake of simplicity, it is proposed here to determine k such that p(1) would be equal to 10 tonnes given the set of parameters already fixed. In first approximation, k can therefore be set to 0.00001. - The result is presented Fig. 1. According to our model, for =0.54, approximately 100, 000 people are required in the colony to achieve full autonomy. Another important result is the peak payload requirement that is close to 800 tonnes and occurs when there are 20, 000 people in the colony. Sending 800 tonnes to Mars each year on average is probably a difficult challenge if there are no commercial exchanges. With current technologies, sending 100 tonnes to low Earth orbit is possible only with the heaviest launchers. With advanced interplanetary propulsion systems and provided that there are no strong time constraints to send the payload, 50 tonnes could probably be sent to the Mars using a single heavy launcher and an ionic propulsion system for the transfer to the other planet. As a consequence, at least 16 launches per year would be required to send 800 tonnes to the colony.

Scenario 2: Frugal society

Robots, modern cars and modern computers are not absolutely required for the survival of the colony. It is indeed possible to live on the planet using 20th century objects. For instance, in the years 1950-1970, there were already cars, planes and even rockets and spacesuits, but computers and robots were rare or very simple. With more simple objects, the productivity is impacted in many domains, but there are huge time savings due to the reduction of the needs: Without smartphones, robots and other complex objects that require miniaturized electronic components, numerous factories would not be needed any more. In addition, simple objects are in general more robust and maintenance is easier. Other working time savings per year are therefore also expected thanks to longer objects lifetimes. On the other hand, a loss in productivity and optimization suggests that the time to mass coefficient would be impacted. For this scenario, r is therefore assumed to be smaller and k higher than in scenario 1. The following parameters are proposed:

-w=2000 hours r = 5x10 5 This value is justified to maintain in first approximation p(1)=10 tonnes (see method in scenario 1). Productivity is less important and miniaturization is less effective. The mass conversion coefficient is therefore higher (assumed twice as high as in scenario 1). Fig. 2. Yearly payload for scenario 2.

The results are presented Fig. 2. Important simplifications of the problem have been made and the interpretation of the results remains a difficult task. However, assuming that the model is not oversimplified, the results suggest that it is preferable to avoid robots and other complex objects to reduce the period needed to achieve autonomy and also to minimize the peak payload requirements per year. With =0.54, the peak payload mass would be around 300 tonnes per year when the colony reaches 27, 000 settlers. Although still a difficult challenge, this is much better than in scenario 1. If only 6 heavy launches were required per year, the colonization process would be close to affordability by a group of rich countries.

Scenario 3: survival

In this scenario, it is assumed that the number of settlers is fixed and the problem is to look for parameters that allow the survival of the colony, which is determined by p(n)=0. Interestingly, the time to mass conversion has no impact on the problem. The only variables are r and  and the equation is very simple (equ.2).

𝑟 = 2000 * 𝑛 ∝

(2) Fig. 3: survival case, for n=100 settlers.

It is shown in Fig. 3 that the survival of the colony for 100 settlers is possible only if two constraints are satisfied:  a very low total working time requirement per person and year. The values are an order of magnitude lower than in scenario 1 and 2. Is that possible? Perhaps, but probably at the expense of good living conditions: People would probably have only very simple tools and would spend most of their time farming and extracting water and other important resources, with very low human resources for industry, medical acts, administration, etc.  the sharing factor is very high, which means that a significant effort would have to be made to share all existing objects, as it could be expected in a big family or perhaps in a communist society, where all the food, the building, the vehicles and all facilities and devices would be shared among all persons.

Conclusion

Our model is very simple. We did not investigate much how to determine the values of w and s but it seems possible to provide estimates. For w, a bottom up approach is possible. The idea would be to fix an industrial strategy (for instance, all objects have to be made of iron or plastics) and to determine how much working time is required to implement all processes, from ores to object manufacturing, then to add the working time for life support, farming, education, medical acts, etc. For the sharing factor, it is also possible to list all objects and consumables and to determine for each of them if it is shared or not, and among how many settlers. A difficult question is how to take into account technological evolutions, development strategies, culture and human factors. Even if the model is very simple, it allows the discussion of different options with different parameters and provides conceptual insights into the difficult question of the feasibility. It is also important to notice that the model does not consider genetic constraints. Some authors tried to determine the minimum number of persons for survival according to the genetic variability [START_REF] Marin | Computing the minimal crew for a multi-generational space journey towards Proxima Centauri b[END_REF]. It is suggested here that the strongest constraint is rather linked to the working time and the available human resources.

Fig. 1 .

 1 Fig. 1. Yearly payload for scenario 1.

  hours It means that 500,000 hours of work would be required to produce all IAC-19-A5,2,7,x49674 Page 4 of 5 objects and consumables needed for a single person per year. s(n) = n  Same role as scenario 1. k = 0.00002

th International Astronautical Congress (IAC), Washington D.C., United States, 21-25 October 2019. Copyright 2019 by Prof. Jean-Marc Salotti.Published by the IAF, with permission and released to the IAF to publish in all form


Management of prosthetic joints infection in France: a national audit


Armelle Cessou
email: armelle.cessou@coria.fr

Florestan Guichard

Hassan Ajrouche

Amath Lo

D single-shot thermography by Spontaneous Raman Scattering in turbulent, spray or oxyfuel flames

Spontaneous Raman scattering noise, limited for thermography in turbulent flames where high spatial and time resolutions are required, is revisited for simultaneous temperature and multispecies concentration single-shot linewise measurements, offering new applications.

Introduction

High temperature generated by combustion processes has a direct impact on a variety of factors including process efficiency, pollutant levels and product quality. It is therefore of great importance to determine temperature with high accuracy, and to provide reliable data for combustion models, which play an increasingly important role in design and optimization procedures. In turbulent flames, few temperature measurement technics are available and reliable in a large range of operational conditions. Raman/Rayleigh measurements are well known to be a powerful diagnostic tool for instantaneous temperature and major species concentration measurements [START_REF] Barlow | Laser diagnostics and their interplay with computations to understand turbulent combustion[END_REF][START_REF] Fuest | Raman/Rayleigh scattering and CO-LIF measurements in laminar and turbulent jet flames of dimethyl ether[END_REF]. Rayleigh scattering is used for temperature measurement, due to its high level of signal-to-noise ratio, associated in an iterative procedure to Spontaneous Raman Scattering (SRS) for major species concentration measurements. By probing of the complete rotational-vibrational energy distribution function with a single laser, SRS is also theoretically a powerful diagnostic for measuring temperature in flames. However its use is restricted by the weakness of its radiation especially in low density media, like flames. Ramanography for time-averaged temperature measurements in flames was proposed by long years ago [START_REF] Lapp | Raman Scattering From Flames[END_REF], and if different procedures adapted to the low level signal were proposed [START_REF] Laurendeau | Temperature-Measurements By Light-Scattering Methods[END_REF], their precision and accuracy for single-shot measurements remains limited, especially when the population of the upper vibrational levels is low. If exposure time in laminar flames can be long enough to accumulate sufficient signal with reduced noise, the investigation of turbulent flames imposed very small spatial and time resolutions, which limit the applicability of SRS thermography. Furthermore combustion community has a strong interest in extending instantaneous temperature-concentration measurements, in more complex configuration as spray flames where Rayleigh scattering measurements are biased by the presence of droplets. Only few works mentioned ramanography in spray flames [START_REF] Grünefeld | Spatially resolved Raman scattering for multi-species and temperature analysis in technically applied combustion systems: Spray flame and four-cylinder in-line engine[END_REF][START_REF] Masri | Raman-Rayleigh Scattering Measurements in Reacting and Non-Reacting Dilute Two-Phase Flows[END_REF]: Masri et al [START_REF] Masri | Raman-Rayleigh Scattering Measurements in Reacting and Non-Reacting Dilute Two-Phase Flows[END_REF] determined temperature using the sum of the species number densities of the major species, Grünefeld et al [START_REF] Grünefeld | Spatially resolved Raman scattering for multi-species and temperature analysis in technically applied combustion systems: Spray flame and four-cylinder in-line engine[END_REF] determined average temperature from Stokes/Antistokes ratio in IC engine in the combustion and exhaust strokes when droplets are absent but in hostile environments.

Providing the complete vibrational distribution function and the temperature, SRS has been shown to be an attractive tool for investigation of non-equilibrium reactive flows, like nanosecond pulsed discharges [START_REF] Lo | Spontaneous Raman scattering: a useful tool for investigating the afterglow of nanosecond scale discharges in air[END_REF][START_REF] Lo | Space and time analysis of the nanosecond scale discharges in atmospheric pressure air: I. Gas temperature and vibrational distribution function of N 2 and O 2[END_REF][START_REF] Lo | Nanosecond pulsed discharge in a propane-air mixture: Ignition and energy deposition[END_REF]. The vibrational distribution function is determined by adjusting theoretical spectra associated to the use of recent spectroscopic data to take into account of the high level of vibrational excitation (up to v=16), and to determination of the apparatus function at each probe volume simultaneously probed along a line. Taking advantage of the recent technological developments in lasers and detectors, the present paper revisits SRS for instantaneous temperature and multispecies concentration measurements in flames. Sufficient signal-to-noise ratio is obtained for instantaneous temperature measurements with an accuracy of 1% and a precision smaller than 3 % in burnt gas of air flames. The high accuracy and precision obtained, show that SRS can be envisaged as an alternative way to Rayleigh-SRS method when high stray light occurs: in two-phase media or near walls. Raman thermography is here assessed for instantaneous measurements in flames, including spray or oxyfuel flames.

Experimental set-up and post-processing

The laser source consists in a Nd:YAG laser (Agilite Continuum) operating at 10 Hz providing up to 1.8 J with tophat pulse with a long pulse duration adjustable from 200 ns till 1 µs. The laser beam is focused using a convergent lens with a 1 m focal-length providing a probe volume of 170 µm-thick (1/e 2 ). The SRS light is collected at 90° with a large solid angle (f/2). The scattered light at the laser wavelength is suppressed by a notch filter (NF03-532E, Semrock). Then, a periscope is used to rotate the image of the laser beam parallel to the entrance slit of the spectrograph (IsoPlane SCT 320, Princeton Instruments) equipped with a grating of 600 grooves/mm and providing a spectral resolution of 0.67 nm. The detector camera was a full-frame back-illuminated CCD (Pixis 400B, Princeton Instruments). The line-wise measurements were performed by imaging almost 3 mm of laser beam, divided in 8 local probe volumes of 50 pixels selected by hardware binning, leading to a spatial resolution of 330 µm.

The collection was gated by a fast electro-optical shutter composed of a Pockels Cell Shutter (PCS) [START_REF] Ajrouche | Assessment of a fast electro-optical shutter for 1D spontaneous Raman scattering in flames[END_REF] to limit flame emission background. With the PCS switched on, the flame emission was integrated during a small time interval (500 ns or 1 µs) chosen to fit the laser pulse duration.

Temperature was determined by the procedure proposed by Lo et al [START_REF] Lo | Nanosecond pulsed discharge in a propane-air mixture: Ignition and energy deposition[END_REF] where the theoretical spectra were convoluted with the in-situ apparatus function and automatically least-square fitted to the experimental spectra.

Results

Bunsen burner flames are investigated fed with methane-air mixtures, ethanol vapor-air loaded with ethanol droplets, or CH 4 -CO 2 -O 2 mixtures. Photograph in Fig. 1 shows the intense Mie scattering of droplets, making impossible temperature measurement by Rayleigh scattering in such a configuration. Figure 1-right illustrates the good signal-to-noise ratio of N2 instantaneous spectra, allowing a theoretical spectrum to be correctly adjusted and the temperature to be determined. In gas flames, the method proposed provides accurate temperature measurements of gas flame with an accuracy of 25 K at 2100 K, and statistical uncertainties around 3% for single-shot measurements with a SNR of 30 for the N 2 peak, which is the parameter affecting in first order the uncertainty.

Figure 1 Line-wise measurements require a high accuracy along the complete probe volume, while the apparatus function can be affected by the location considering the short focal length of the collection lens and the moderate length of the spectrograph. The procedure to determine the apparatus function at each probe volume leads to a high accuracy of the temperature measurement along the complete length probed (maximum error of 1% in burnt gas) and provides accurate measurements with temperature as low as 500 K, while with low dispersion spectra used for multispecies measurements the minimum temperature is 800 K when Stokes/Antistokes band ratio is used.

In two-phase flames, the accuracy and precision are identical as in gas flames for a same signal level, despite the strong elastic scattering. However the laser energy was a little reduced to avoid optical breakdown, leading to precision of 4.6% in burnt gas. The method is extended to oxyfuel flames, where N 2 is absent.

This work is funded by the BIOENGINE project co-financed by the European Union (ERDF) and the Normandie Regional Council.


Essodong Barcola

Micky Rakotondrabe

Morvan Ouisse

Ausrine Bartasyte

Electrical design and simulation of kinetic piezoelectric harvester for distributed control cells

Keywords: Piezoelectric energy harvesting, electrical circuit, booster, rectifier, voltage multiplier, oscillator, adaptator

This paper deals with the design, the modeling and the simulation of two electrical circuits devoted to kinetic piezoelectric energy harvesters for powering distributed control cells. Two schemes are proposed, analyzed and discussed: the first is a based on classical electrical stages, and the second integrates a novel multiplier/rectifier stage which permits to reduce the voltage loss. Simulation and experimental tests are carried out and demonstrate the advantage of the second scheme relative to the first one.

INTRODUCTION

This paper is within the context of distributed control cells in [START_REF]Integrated Smart Systems[END_REF] where one cell is considered as an independant mechatronic module with embedded sensing, actuation, control and power source. One of the application of the distributed control cells is the distributed damping of vibrations in a vibrating environment such as in cars chassis. Several projects in link with the above context deal with the sensing, the actuation and the control, for example the C-MUMS project [START_REF]Control of Multivariable Piezoelectric Microsystems with Minimization of Sensors[END_REF] . However this paper is within the project that studies the power source of each cell and in which alternative ways than batteries are studied.

In an environment with ambiant vibrations, it is possible to extract the vibrational energy and transform it into electrical energy. Vibrational piezoelectric energy harvesting (VPEH) is an approach known for that. In fact VPEH has been given increasing attention these last twenty years. This is thanks to the increasing trend towards autonomous miniaturized apparatuses with low energy consumption [START_REF] Agnus | Robotic microassembly and micromanipulation at FEMTO-ST[END_REF][START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF][START_REF] Rakotondrabe | Piezoelectric systems for precise and high dynamic positioning: design, modeling, estimation and control[END_REF] . In a VPEH, the principle consists in scavenging the vibrational energy with a piezoelectric structure and an electrical circuit. The piezoelectric structure transforms the piezoelectric material deformation due to the external vibrational movement into varying electrical voltage, and the circuit conditions this latter into a more exploitable source (regulated and eventually amplified voltage, regulated power).

In order to be able to extract more and more energy from smaller and smaller piezoelectric structures, the research in VPEH can be classified into five categories. The first category studies the possibility to increase the piezoelectric layers within the transducer structure [START_REF] Rabenorosoa | Performances analysis of piezoelectric cantilever based energy harvester devoted to mesoscale intra-body robot[END_REF][START_REF] Rakotondrabe | Towards high autonomy energy harvesters based on piezoelectric MEMS[END_REF][START_REF] Song | Multilayer piezoelectric energy scavenger for large current generation[END_REF][START_REF] Abramovich | Multi-layer piezoelectric generator[END_REF][START_REF] Zhu | Improving Output Power of Piezoelectric Energy Harvesters using Multilayer Structures[END_REF] . If the layers number is increased, the recuperated voltage is increased. A challenge in this category is the difficulty to realize the multilayered structure when the expected total thickness is less than the millimeter. Recent research shows that such realization could be possible if performed with clean-room processes [START_REF] Bienaimé | Static / dynamic trade-off performance of PZT thickfilm micro-actuators[END_REF] . The second category of research in VPEH deals with combining piezoelectric transduction with another type of transduction [START_REF] Yang | Hybrid energy harvester based on piezoelectric and electromagnetic mechanisms[END_REF][START_REF] Shan | A new energy harvester using a piezoelectric and suspension electromagnetic mechanism[END_REF][START_REF] Lee | Hybrid energy harvester based on nanopillar solar cells and PVDF nanogenerator[END_REF][START_REF] Gambier | Piezoelectric, solar and thermal energy harvesting for hybrid low-power generator systems with thin-film batteries[END_REF] . Called hybrid PEH, the approach permits to obtain larger power since the total output power is a combination of that from the PEH with that of the second transducer which can be thermal, magnetic, solar. Nowadays, a limitation of hybrid PEH is the size: the total dimensions of the harvester increase with the number of transducers. Another category of VPEH research is focused on how to make the harvester sensitive to different vibrations frequency (broadband harvesters) [START_REF] Eichhorn | A frequency tunable piezoelectric energy converter based on a cantilever beam[END_REF][START_REF] Lallart | Frequency self-tuning scheme for broadband vibration energy harvesting[END_REF][START_REF] Challa | Towards an autonomous self-tuning vibration energy harvesting device for wireless sensor network applications[END_REF][START_REF] Tang | Broadband vibration energy harvesting techniques[END_REF] .

Because the output power is directly related to the frequency of the vibrations, classical VPEH are only efficient for a given frequency. In this category therefore, the objective is to develop harvesters utilizable at various excitation frequency. The fourth category in VPEH research focuses on the improvement of the piezoelectric material itself [START_REF] Nemesis | Novel Energy Materials: Engineering Science and Integrated Systems[END_REF][START_REF] Harvest | Flexible nanowire devices for energy harvesting[END_REF][START_REF] Nanogen | Polymer-based piezoelectric nanogenerators for energy harvesting[END_REF] . In particular, Lithium Niobate materials are shown to be a good alternative material face to lead-zirconate-titanate (PZT) [START_REF] Bartasyte | From green piezoelectric materials to designed hybrid piezoelectric energy harvesters[END_REF][START_REF] Bartasyte | Can LiNb03 be an alternative for PZT in vibrational energy harvesters?[END_REF] . They are lead-free materials and could provide performances that concurrent those of PZT. Finally, the last but not the least, the research on the electrical circuit itself constitutes a category in VPEH research. The endeavor is to develop circuits with less consumption, more efficient energy transfer and conversion as well as more efficient voltage adaptation and regulation [START_REF] Dallago | Electronic interface for piezoelectric energy scavenging system[END_REF][START_REF] Guyomar | Recent progress in piezoelectric conversion and energy harvesting using nonlinear electronic interfaces and issues in small scale implementation[END_REF][START_REF] Lallart | Small-scale energy harvesting[END_REF][START_REF] Lallart | Switching delay effects on nonlinear piezoelectric energy harvesting techniques[END_REF] . This paper lies in the last category. It presents our first studies on electrical circuit for VPEH devoted to one control cell. The first contribution of the paper is the suggestion of an electrical scheme on the basis of combination of classical electrical stages. In the second contribution, we introduce a new stage that serves as voltage multiplier and rectifier simultaneously. The two electrical schemes are discussed, simulated and tested. For that, the paper is organized as follows. Section-2 presents the first electrical circuit. Both simulation and experimental results are presented in the same section. In section-3, the suggested second circuit is detailed. Again, both simulation and experimental results are presented. The last section deals with discussions, concluding remarks and future works.

AN ELECTRICAL CIRCUIT BASED ON STANDARD STAGES

Generally, the output signals of a piezoelectric structure in a VPEH are not favourable. The output voltage is alternating and could be of high amplitude, the output current is low (and so is the power), and the output impedence is capacitive [START_REF] Badel | Récupération dénergie et contrle vibratoire par éléments piézoélectriques suivant une approche non linéaire[END_REF] . It is therefore important to adapt the output signals with respect to the requirements and with respect to the load, where constant and regulated voltages are often required. The basic and classical electrical circuit for a PVEH is a four diodes based rectifier with a smoothing and filtering capacitor [START_REF] Ertuck | Piezoelectric energy harvesting[END_REF] . This simple technique has several limitations. First when the piezoelectric transducer furnishes low amplitude alternating votage, the diodes absorb it and there is not anymore exploitable voltage at the output of the circuit. If the voltage amplitude is too low, the diodes may not even work. Additionally to that, if there is a variation of the vibrations amplitude which directly affects the alternating voltage of the piezoelectric transducer, the output DC voltage also varies and is not regulated. Finally, with this simple circuit, the output DC voltage itself and the output power depend on the load. In this section, we suggest an electrical scheme based on existing stages in the literature in order to bypass some of these limitations.

The electrical scheme

Let Fig. 1 depicts the whole block-diagram of the proposed electrical circuit and the piezoelectric transducer. The piezoelectric structure transducer is excited by external vibrations at a frequency f o and oscillates with an amplitude δ. Thanks to the direct piezoelectric effect, charge Q appears at its electrodes. Therefore, due to the internal capacitor of the piezoelectric element, an alternating voltage u p also appears accross it. Following the technique suggested in [START_REF] Badel | Récupération dénergie et contrle vibratoire par éléments piézoélectriques suivant une approche non linéaire[END_REF] and references herein, we suggest to add a booster in order to amplify the amplitude of u p such that there is s sufficient voltage to run the rectifier. Then the alternating output u b is transformed into DC voltage u rect with a rectifier and a smoothing capacitor stage. Finally, the last stage deals with transforming the DC voltage u rect into the required votage u, either by multiplying or dividing it, by regulating this latter, and eventually by adapting the output impedence. Specifically the impedence adaptor can be designed to disconnect the load from the upstream stages, so that the output power and output voltage become independent from the vibrations frequency f o . Fig. 2 depicts the booster, the rectifier and the smoothing capacitor stages. The booster is a nonlinear circuit composed of an inductance and two switches. Called SSHI (synchronized switch harvesting in inductor) [START_REF] Badel | Récupération dénergie et contrle vibratoire par éléments piézoélectriques suivant une approche non linéaire[END_REF] , its principle consists in detecting the extremum (resp. positive or negative) and, when the extremum is detected, closing the switch thanks to the nonlinear circuit (resp. positive alternance switch or negative alternance switch) during a very few delay relative to the period 1 fo . By doing so, the output voltage u b is a combination of the voltage from the inductor and that from the piezoelectric transducer. Other schemes of booster can be found in the literature, for instance in [START_REF] Badel | Récupération dénergie et contrle vibratoire par éléments piézoélectriques suivant une approche non linéaire[END_REF] and references herein. They are principally nonlinear circuits. Finally, as depicted in Fig. 2, the suggested rectifier is based on a four diodes bridge. Without considering the impedence adaptor, let Fig. 3 depicts the suggested voltage multiplier. In order to multiply the DC voltage u rect however, a rectangular (or square) signal is first required. We therefore add an oscillator as shown in the figure. Since the input of the oscillator is a DC voltage, it has the same objective than an electronic chopper. Then according to the wanted multiplication factor, a series of diode and capacitor systems is proposed for the multiplier itself. Finally, if an impedence adaptor is required, it can be placed before the oscillator or after the multiplier. It worthy that the output power and output voltage depend on the frequency f o of the oscillating piezoelectric element, and consequently on the frequency of the alternating voltage u p . However, in our case, the adaptor can be designed so that the load to which is applied u is virtually disconnected from these upstream stages, and consequently u is independant from the frequency f o . It is one of the significances of the proposed scheme in Fig. 1.

Simulation

The electrical circuit suggested in Fig. 1 and detailed in Fig. 2 and Fig. 3 has been simulated using LTSPICE software. During the simulation, a low frequency voltage generator is used rather than a piezoelectric circuit, this latter being non-implementable in the software. All the diodes used are based on the 1N4148 type, the NPN transistors on the 2N2222, the PNP transistors on the 2N3906 and the MOSFET transistors on the IRFZ44N. All the simulation and experimental tests were carried out with low frequency f o because the application context is expected to work at low frequency vibrational environments.

First, a sine input voltage u p with amplitude of 1V and a frequency of 1Hz has been applied. Fig. 4-a shows the evolution of the output voltage u over the time. The figure shows that the voltage transient part takes several tens of seconds before reaching the nominal voltage. Furthermore, the output nominal voltage is less than the input applied voltage, i.e. less than 1V , and the total gain between the amplitude of u p and the output voltage u is less than 1. This is due to voltage loss absorbed by the components used in the electrical circuit. In particular, each diode requires an amount of voltage when conducting. Now, we simulate the circuit with an input voltage u p having an amplitude of 2V . Fig. 4-b shows the results. It shows that the output voltage largely exceeds 3V when observed for several hundreds of seconds. Consequently the voltage gain is almost six times higher than the previous test. In fact, the voltage loss in the different components has been covered and thus the circuit starts to work correctly and to amplify the voltage. The circuit has also been simulated at higher frequencies but similar results and remarks are obtained.

Experimental results

The circuit in Fig. 1 has been realized with the same components used for the simulation. A piezoelectric cantilever structure commercially available was first tested as transducer to supply u p . Due to the very low voltage of the piezoelectric transducer (tens of milliVolts) which is insufficient to make the components start working, we finally used a voltage generator to supply u p . First, a sine voltage u p with an amplitude of 1V and frequency of 1Hz is tested. Contrary to the simulation results which showed a reduced output voltage (u ≈ 500mV ), the experiments provide an output u of about 2.5V . The reason of this difference between the simulation and the experimental results might be the voltage consumption in the simulated circuit which could differ from that of the realized circuit. Then, the same amplitude (1V ) of input but with higher frequency (10Hz) has been tested. We obtain an output u of about 3V as depicted in Fig. 5 in which the output voltage u.

A NEW VERSION OF ELECTRICAL CIRCUIT

The new scheme

In the previous scheme (Fig. 1), the booster increases the alternating and sine voltage u p . It is shown that the output voltage u b of the booster is not anymore sine but is closer to a modified rectangular signal [START_REF] Badel | Récupération dénergie et contrle vibratoire par éléments piézoélectriques suivant une approche non linéaire[END_REF] . The objective of the new version of electrical circuit in this section is to directly use this signal as the input of a voltage multiplier, which normally requires a signal from an oscillator. The advantage is that we remove the rectifier stage and the oscillator sub-stage and consequently we reduce the consumption and the voltage loss in the whole circuit. If required however, another adaptor/multiplier stage can be added before the load in order to obtain higher output voltage.

Let Fig. 6-a depicts the suggested second version of electrical circuit. The booster is the same than that of Fig. 1. Its output u b is not anymore a sine but has a shape between a sine and a rectangular signal. This signal is afterwards directly employed as the input of a multiplier, no oscillator being required anymore. However, because u b is not a pure rectangular signal, we suggest to slightly modify the multiplier in order to include a rectifier part such that a DC output u m is ensured. The suggested integrated multiplier/rectifier stage is depicted in Fig. 6-b and is based on a series of diodes and capacitors. Finally, an adaptor/multiplier stage can be again added to increase the DC output u m as well as to adapt the impedence. Without considering the adaptor, the voltage multiplier here is again composed of an oscillator (to provide rectangular signal) and of the multiplier itself, similarly to Fig. 3.

Simulation results

The electrical scheme in Fig. 6 was implemented in LTSPICE and simulated. First, all the capacitors in the multiplier/rectifier were set equal to 10µF . A sine input voltage u p with an amplitude of 1V and a frequency of 1Hz is applied to the whole circuit. Fig. 7-a shows the results. They show that the output voltage u exceeds 1.2V , which is more than two times higher than that of the first circuit in Fig. 1. This gain is explained as follows: instead of consuming voltage due to the four diodes based rectifier (see Fig. 1), here we amplify the voltage thanks to the multiplier/rectifier. We now set the capacitors in the multiplier/rectifier equal to 1000µF . The simulation results are depicted in Fig. 7-b and show an output voltage tending towards 14V , which is almost thirty times higher than that obtained with the first circuit in Fig. 1.

Experimental results

The electrical scheme in Fig. 6 has been realized. The used diodes and transistors are similar to those for the first circuit. We use capacitors of 1µm for the multiplier/rectifier. Again, a piezoelectric cantilever structure commercially available was first tested as transducer. Unfortunately its output voltage u p (in the order of tens of milliVolts) was not sufficient to trigger the electrical circuit. Hence, we use a generator to supply u p and to verify the functioning of the electrical circuit. Fig. 8 shows the results when an input voltage u p with an amplitude of 1V and a frequency of 1Hz is applied. The output u reaches 3V which is much higher than the simulation. Again, the reason might be that the voltage consumption in the simulated circuit could differ from that of the realized circuit.

DISCUSSIONS, CONCLUSIONS AND FUTURE WORKS

In this paper, we suggested two electrical circuits for VPEH devoted to supply devices in distributed control cells. The first suggested circuit was based on already existing electrical stages which we suggested to combine: a booster stage, a rectifier with smoothing capacitor stage and an adaptor/multiplier stage. In the second circuit we suggested to replace the rectifier by a multipier which, instead of having a votage consumption stage, the voltage is increased. Different simulation and experimental tests were carried out and shown the advantage of the second circuit relative to the first one. The experiments also indicated that there was a minimum input voltage required from the piezoelectric structure in order to make the electrical circuit work correctly. This minimum voltage corresponds to the required consumption of the different electrical components. In both circuit, the multiplier stage can be augmented by an adaptor which permits to virtually disconnect the load to be supplied with the upstream stages. The advantage is that the output voltage and output power can be independant from the load.

The components (diodes, transistors) used during the simulation and experimental tests in this paper were standard. The next work consists in replacing all the components by low-consumption components in order to be able to work with low voltage VPEH.

Figure 1 :

 1 Figure 1: Block diagram of the electrical circuit.

Figure 2 :

 2 Figure 2: The booster and the rectifier.

Figure 3 :Figure 4 :

 34 Figure 3: The voltage multiplier.

Figure 5 :

 5 Figure 5: Experimental results: output voltage u with an input voltage u p = 1V × sin(1 × π × 10Hz × t).

Figure 6 :

 6 Figure 6: (a): block diagram of the second electrical circuit. (b): scheme of the multiplier/rectifier stage.

Figure 7 :

 7 Figure 7: Simulation results with an input voltage u p = 1V × sin(2 × π × 1Hz × t). (a): with capacitors of 10µF in the multiplier/rectifier. (b): with capacitor of 1000µF in the multiplier/rectifier.

Figure 8 :

 8 Figure 8: Experimental results: output voltage u with an input voltage u p = 1V × sin(1 × π × 10Hz × t) and capacitors of 1µF in the multiplier/rectifier.

ACKNOWLEDGMENTS

This work is supported by the national ANR-JCJC C-MUMS-project (National young investigator project ANR-12-JS03007.01: Control of Multivariable Piezoelectric Microsystems with Minimization of Sensors). This work is also supported by the LABEX ACTION (ANR-11-LABX-0001-01).