Margaux Topenot

Gaël Chevallier

Morvan Ouisse

Damien Vaillant

Model Reduction of Electric Rotors Subjected to PWM Excitation for Structural Dynamics Design

Keywords: Electric motor, vibratory fatigue, Pulse Width Modulation, Structural dynamic design, model reduction

Rotors of asynchronous machines can be subjected to risk of failure due to vibratory fatigue. This is caused by the way electric motors are powered. Pulse Width Modulation (PWM) is the control strategy of the traction chain. This signal is composed by a fundamental and numerous harmonics of voltage and current that induce harmonics on the torque signal resulting in huge torque oscillations. It can lead to repeated torsional resonance when coincidences occur. This can induce severe damages and even lead to rupture if electric excitations are not taken into account at the design stage. In this work, a magnetic finite element model is built by using Fourier decomposition in order to take into account harmonics due to PWM. Pressures exported from this model are used as inputs for mechanical FEM. A mechanical reduced order model is also proposed in order to compute stress in rotating part. This second model allows to reduce time computation and then to consider several operating points to build a complete speed up. A correlation is performed between these two models and rotating tests in order to discuss the relevance of these approaches to design rotor parts.

Introduction and state of the art

Trains are set into motion through electrical energy. From a theoretical point of view, electric machines can be driven by sine wave signals but practically, due to signal synthesis issues and power limitations, motors are driven using Pulse Width Modulation (PWM). This allows to manage easily the variation of the rotating speed. The PWM is based on a fundamental frequency equal to the sine signal frequency. However, PWM generates small-amplitude oscillations at frequencies corresponding to the harmonics, inducing vibrations and spurious mechanical stress. Figure 1 illustrates the difference between PWM and sine waveform signals. This results in torque oscillations over a wide frequency range and possibly large amplitudes, sometimes leading to the rupture of several rotating parts due to resonance [START_REF] Bruzzese | Failure root-cause analysis of end-ring torsional resonances and bar breakages in fabricated-cage induction motors[END_REF].

Wachel plots interference diagram in 1993 in order to check for coincidence of the torsional mode with critical operating speed [START_REF] Wachel | Analysis of Torsional Vibrations in Rotating Machinery[END_REF]. Song-Manguelle proves in [START_REF] Song-Manguelle | Prediction of Mechanical Shaft Failures Due to Pulsating Torques of Variable-Frequency Drives[END_REF] and Feese in [START_REF] Feese | Torsional vibration problem with motor/ID fan system due to PWM variable frequency drive[END_REF] that the variable frequency drive leads to shaft failures due to pulsating torque. Other rotating parts can be subjected to failures like fans [5] [6]. The authors of [START_REF] Kerkman | PWM inverters producing torsional components in AC motors[END_REF] recommend to pay attention of the drive's voltage spectrum to design mechanical parts. Electromagnetic and mechanical phenomenon should be taken into account, this is why several authors propose to couple the two physics : Hallal and Pellerey propose a weakly coupled methodology in [START_REF]Études des vibrations d'origine électromagnétique d'une machine électrique: conception optimisée et variabilité du comportement vibratoire[END_REF] and [START_REF] Pellerey | Coupled numerical simulation between electromagnetic and structural models. Influence of the supply harmonics for synchronous machine vibrations[END_REF]. As Hallal, Dupont proposes in [START_REF] Dupont | Vibroacoustic simulation of an electric motor methodology and focus on the structural FEM representativity[END_REF] a methodology to project magnetic load on mechanical mesh. Journeaux exposes in [START_REF] Journeaux | Modélisation multi-physique en génie électrique[END_REF] the advantages and the drawbacks of weak and strong coupling. Delforge alerts on the huge computational time to perform such a study in [START_REF] Delforge | Induction machine modeling using finite element and permeance network methods[END_REF]. Van der Giet and al. make the comparison between 2D and 3D coupled electromagnetic and structural dynamic simulation in [START_REF] Van Der Giet | Comparison of 2-D and 3-D Coupled Electromagnetic and Structure-Dynamic Simulation of Electrical Machines[END_REF]. From the mechanical point of view, the rotor is a complex assembly. Its modeling is subjected to various assumptions. In particular, the stack lamination is relatively hard to model. Millithaler proposes in [START_REF] Millithaler | Structural dynamics of electric machine stators: Modelling guidelines and identification of three-dimensional equivalent material properties for multi-layered orthotropic laminates[END_REF] a methodology to model laminated structures. An identification methodology is proposed by Mogenier in [START_REF] Mogenier | Identification et prévision du comportement dynamique des rotors feuilletés en flexion[END_REF] to get material parameters in rotor subjected to bending modes.

The aim of this study is to propose and compare rotors dynamic models able to capture the resonance phenomenon due to PWM excitation. Two models are proposed: the first one consists in weakly coupling magnetic and mechanical finite elements models by using Fourier expansion in order to take into account harmonics due to PWM whereas the second one uses reduced order modelling in order to obtain the structural response faster. The scale of refinement which is necessary to describe the problem with a good accuracy will be discussed after the presentation of the results. Rotating tests have been performed in order to measure the strain in rotating parts for a fixe speed that corresponds to the resonance speed. Figure 2 shows a block diagram that explains how models is compared to reference data.

PWM synthesis and stress analysis

Electro-technical engineers are responsible for the PWM synthesis in the inverter in order to power trains. Several strategies are availables for signal generation [START_REF] Bowes | Computer-aided design of PWM inverter systems[END_REF] according to industrial requirements. This mission requires to wisely choose the way that impulses are created by avoiding losses and increasing performance. It can directly impact the traction chain but also the dynamic behavior of electric rotors. Among all the strategies, asynchronous, synchronous and space vector modulation are the most common. For the two first one, carrier signal is compared to a reference sine waveform signal. This comparison leads to the opening or the closing of switches (IGBT in the inverter). The difference between both lies in the ratio of carrier frequency and the sine frequency. Space vector modulation is a really interesting way to get PWM signals. Indeed, this strategy allows to choose the switching angles (corresponding to switching times). This means that it is possible to change harmonics frequencies in the spectrum of the signal. Due to the performance requirements, it is difficult to completly avoid harmonics but a good knowledge of the studied structure can help to avoid structural dynamic problems.

In this paper, the electromagnetic part of the motor is modeled by an equivalent electric circuit with voltage drive. The voltage source corresponds to the sum of the harmonics of the PWM signals by using Fourier expansion. Authors alert on the limitation of the number of variables that can be set and therefore on the limitation of the number of harmonics taken into account in commercial calculation codes. In general, the carrier frequency is really high compared to frequencies of interest for structural dynamic design. It means that filters can be applied to lower the number of harmonics, but the user must be aware that the reconstructed signal is changed. In addition, if no filter is applied, the maximal frequency is the carrier frequency and then it could lead to very small time steps and then to huge computational time and results files.

A 3D mechanical finite element model of an electric rotor is considered as reference. The torque provided by the electromagnetic simulation is applied at the rotor center and is also used as load for a reduced order model (see figure 2). [START_REF] Bruzzese | Electromechanical modeling of a railway induction drive prone to cage vibration failures[END_REF]. He managed to define an electrotechnical model by solving equivalent electrical circuit and applying magnetic torque on a simplified mechanical model composed by masses, springs and dampers. Inertia and stiffness matrices were evaluated from a lumped representation. Here, the mechanical reduced order model is built from the modal basis and nodal coordinates of sections coming from the 3D FEM in order to have a more accurate mechanical representation of the real rotor compared to the work proposed in [START_REF] Bruzzese | Electromechanical modeling of a railway induction drive prone to cage vibration failures[END_REF]. The key of such a model is the transformation matrix that allows the reduction. Starting from a full 3D model composed of thousands of degrees of freedom, the model is reduced to only nine rotational degrees of freedom. This work being focused on torsion behavior of the electric machine, only rotation degrees of freedom around motor axis are considered. The rotor is decomposed in nine sections as shown in figure 3. The reduction matrix is built by modal projection using the link between displacement and rotational degrees of freedom of rotor sections.

{ q} + [∆]{ q} + [Λ]{q} = { f } (1)
with

[∆] =    . . . 2ξ i ω i . . .   , ξ i being the modal damping ratio, [Λ] = [Φ] T [K][Φ] =    . . . ω 2 i . . .   , [Φ]
being the modal basis and [K] the stiffness matrix, { f } = [Φ] T {F}, {F} being the load vector. The mechanical stresses are obtained by using the modal stresses :

{σ } = [Φ σ ]{q}. ( 2
)
3 Results

Load excitations and modal behavior should be mastered to predict the dynamic response of the rotor. Figure 2 shows what quantities should be compared. The torque signal and the spectrum exported from electromagnetic simulation is illustrated in figure 4. This is the input load for mechanical computation. Numerically, high torque harmonics are close to the natural frequency of the rotor. But experimentaly, the torque has been measured with a device that allows only to measure a signal until 250Hz. This means that torque spectrums can not be compared. However, mean values of experimental and numerical torque can be compared in order to validate the simulation methodology. The 5% gap between the two indicates that the model is representative of the test conditions.

A coincidence between the load harmonics and the natural frequency of the torsion mode is observed in the test data. In order to reproduce this effect with models, the maximal torque value in the spectrum has been spread around the torsional eigen frequency in numerical models. This spreading is necessary because a small error in models can cause the absence of the coincidence. It results in 4% of relative error on the maximal stress at the torsional resonance between the full model and the reduced order model. This indicates that reduced order model is efficient to describe the behavior of the structure and allows to consider a full speed up.

The computation of the modal basis takes 8 minutes on a calculator with 16 CPU, 3.2GHz and 225692 MB RAM and then the calculation of rotation and stresses thanks to reduced model takes 2 minutes. In comparison, the magnetic computation with magnetic solver takes 5 hours on a laptop with 8 CPU, 2.7GHz and 16 Go RAM and then the mechanical computation with mechanical FE solver takes 2 hours on the calculator.

The figure 5 illustrates qualitatively the comparison of the measured waterfall of the stress (left side) and the computed one (right side). Resonance clearly appears when load harmonics cross the eigen frequency.

It remains to calibrate the model to fit the rotating test data. Several aspects should be taken into account :

• the damping ratio of the rotor has been measured in free-free condition and may be very different in the full motor in rotating condition ;

• even if the constant value of the computed torque fits well the average value of the experimental mechanical torque, the computed torque is the electromagnetic torque and due to losses, it might not be identical to the mechanical torque. The inertia of the rotor might filter the torque ; • uncertainties about the location and the orientation of the gauges can lead to a bad correlation between measured and computed strain ;

• during tests, the speed is oscillating and it can lead to a rotor that is not excited exactly at the resonance, or the slip changes during tests due to temperature that not allows to stay at the resonance for a long time.

All these aspects can explain differences between tests and models. Modelling assumptions must be carefully chosen.

Conclusion

This paper compares two computational methodologies to investigate the vibratory response of electric rotors subjected to PWM excitation. The first one combines two Finite Elements Models in order to solve the multiphysic problem. The torque computed from the magnetic solver with test PWM signal is the input of the computation of the structural response to determine the mechanical stresses. Secondly, a model reduction is proposed to fasten the computation time during design phase. The 4% difference on maximal stress at the torsional resonance between the full model and the reduced order model allows to conclude that the reduction is accurate enough and efficient to compute the stress along a speed up. However, it remains to calibrate models to tests data. Future work will focus on the understanding of phenomenon and assumptions to be choose to fit tests data. Model updating will allow a better prediction of rotor vibratory response.

Figure 1 :

 1 Figure 1: Electric current intensity waveform with PWM or sine

Figure 2 :

 2 Figure 2: Block diagram illustrating the methodology

Figure 3 :

 3 Figure 3: Rotor divided in 9 sections

Figure 4 :

 4 Figure 4: Torque oscillations due to PWM (left side) and FFT of this signal (right side)

Figure 5 :

 5 Figure 5: Comparison of stress waterfall obtained by measurement (left side) and by computation with the reduced order model (right side)


Ting-Ting Wang

Sylwester Bargiel

Franck Lardet-Vieudrin

Yan-Feng Wang
email: wangyanfeng@bjtu.edu.cn

Yue-Sheng Wang

Vincent Laude
email: vincent.laude@femto-st.fr

Collective resonances of a chain of coupled phononic micro-resonators

We study experimentally a chain of defect resonators in a phononic crystal slab and observe its collective resonances at ultrasonic frequencies of a few MHz. A phononic crystal of cross-holes is fabricated in a thin fused silica plate by femtosecond laser writing followed by KOH etching. A chain of 17 coupled resonators is defined with no definite spatial periodicity but similar coupling strength between nearest neighbor resonators. The full phononic band gap ensures that there are only evanescent waves in the crystal which can tunnel between adjacent resonators in the plane. The resulting evanescent coupling strength decreases exponentially with distance. Collective resonances are excited by a frequency-driven piezoelectric vibrator attached at one end of the chain and imaged along the chain with a laser Doppler vibrometer. A discrete spectrum of resonances is observed and explained by a model representing the chain as a phononic polymer. The theoretical analysis is supported by finite element simulations that agree well with experimental results. Chains of evanescently-coupled micro-resonators forming phononic polymers could find applications for ultrasonic sensing, for implementation in topological phononics, and for the design of optomechanical resonator chains.

I. INTRODUCTION

Over the past two decades, a great deal of effort has been made to achieve control of waves propagating within phononic/sonic crystals exhibiting elastic/acoustic band gaps [START_REF] Kushwaha | Acoustic band structure of periodic elastic composites[END_REF] . The propagation of waves is completely forbidden in band gaps [START_REF] Sigalas | Band structure of elastic waves in two dimensional systems[END_REF] . Actually, even though there are no waves propagating inside a band gap of a perfect crystal, evanescent waves still exist [START_REF] Laude | Phononic crystals: artificial crystals for sonic, acoustic, and elastic waves[END_REF] . Besides explaining complex solutions in the dispersion relation of homogeneous media and crystals [START_REF] Laude | Evanescent Bloch waves and the complex band structure of phononic crystals[END_REF] , evanescent waves also provide a coupling mechanism for the vibrations of periodic chains of masses connected by spring [START_REF] Jensen | Phononic band gaps and vibrations in oneand two-dimensional mass-spring structures[END_REF][START_REF] An | Elastic wave and vibration bandgaps in two-dimensional acoustic metamaterials with resonators and disorders[END_REF] or of coupled-resonator waveguides [START_REF] Wang | Evanescent-wave tuning of a locally resonant sonic crystal[END_REF][START_REF] Wang | Coupling of evanescent and propagating guided modes in locally resonant phononic crystals[END_REF] . Strong confinement and localization of waves can indeed be achieved at the frequencies in the complete phononic band gap. [START_REF] Torres | Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects[END_REF][START_REF] Pennec | Two-dimensional phononic crystals: Examples and applications[END_REF][START_REF] Akahane | Design of a channel drop filter by using a donortype cavity with high-quality factor in a two-dimensional photonic crystal slab[END_REF] . When the periodicity of the perfect crystal is broken, for instance by changing the geometry of the unit cell or the material properties, point and line defects can form cavities and waveguides, respectively [START_REF] Wang | Tunable fluid-filled phononic metastrip[END_REF][START_REF] Jin | Tunable waveguide and cavity in a phononic crystal plate by controlling whispering-gallery modes in hollow pillars[END_REF] .

Waveguides composed of linear chains of coupled resonators in a crystal were first proposed in the field of photonics [START_REF] Yariv | Coupledresonator optical waveguide: a proposal and analysis[END_REF] . These coupled-resonator waveguides can be designed using photonic-crystal defect cavities or micro-ring resonators [START_REF] Notomi | Large-scale arrays of ultrahigh-Q coupled nanocavities[END_REF] , and can have applications to slow light [START_REF] Baba | Slow light in photonic crystals[END_REF] , light storage [START_REF] Xia | Ultracompact optical buffers on a silicon chip[END_REF] , sensing [START_REF] Wang | Silicon coupledresonator optical-waveguide-based biosensors using lightscattering pattern recognition with pixelized mode-fieldintensity distributions[END_REF] , and light capture [START_REF] Otey | Completely capturing light pulses in a few dynamically tuned microcavities[END_REF] . Dielectric coupled-resonator waveguides also parallel plasmonic waveguides composed of periodically distributed metal nanoparticules [START_REF] Oulton | A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation[END_REF] . Coupled-resonator waveguides have been since then extended to the field of phononic crystals. Khelif et al. demonstrated experimentally the guiding and the bending of acoustic waves in highly confined waveguides [START_REF] Khelif | Guiding and bending of acoustic waves in highly confined phononic crystal waveguides[END_REF] . Hatanaka et al. developed a phonon waveguide using a one-dimensional array of suspended membranes and demonstrated that it could support and guide mechanical vibrations by nano-electro-mechanical systems (NEMS) [START_REF] Hatanaka | Phonon waveguides for electromechanical circuits[END_REF] . Wang et al. investigated experimentally Lamb wave propagation in coupled-resonator elastic waveguides formed by a chain of cavities in a two-dimensional phononic crystal slab with cross holes [START_REF] Wang | Guiding and splitting Lamb waves in coupledresonator elastic waveguides[END_REF] . They also investigated the transmission properties of coupled-resonator acousto-elastic waveguides formed by a chain of cavities with water, such as multiply 90 • -bent waveguides or wave splitters [START_REF] Wang | Reconfigurable phononic-crystal circuits formed by coupled acoustoelastic resonators[END_REF] .

Defects in a phononic crystal have long been known to introduce localized states of vibration [START_REF] Psarobas | Phononic crystals with planar defects[END_REF] . They have been proposed to filter the transmission through a crystal sample [START_REF] Khelif | Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal[END_REF] or to alter the transmission along a waveguide [START_REF] Pennec | Acoustic channel drop tunneling in a phononic crystal[END_REF] . In general, the defects used were small and chosen to possess well defined isolated resonance frequencies. An additional condition of periodicity at the level of a super-cell is generally imposed, in order to guarantee that the dispersion can be obtained numerically via the computation of phononic band structures. In this paper, we instead examine an extended and non periodic defect in a phononic crystal, basing on the coupled-resonator concept, that possesses a discrete spectrum of closelyspaced resonances. We remark that coupling between adjacent resonators is not limited to straight lines or to privileged crystallographic directions. Instead, evanescent coupling inside a complete band gap is omnidirectional and decreases exponentially away from a resonator. Hence, it provides an adequate platform to implement nearest-neighbor coupling within an aperiodic chain of finite length, without the additional propagation phases that would result from attaching the resonators to a propagation medium or a surface [START_REF] Benchabane | Surface-wave coupling to single phononic subwavelength resonators[END_REF][START_REF] Raguin | Coherent interplay between surface acoustic waves and coupled mechanical resonators: transition from plasmon-like to surface mediated coupling[END_REF] . Actually, interaction between adjacent resonators is reminiscent of the chemical bonds of molecular polymers. Thus the chain of resonators is loosely termed a 'phononic polymer' in this work and it is understood that it can be deformed rather freely in the plane. In the following, we specifically consider a chain of coupled resonators fabricated in a phononic crystal slab in fused silica that can be conveniently manufactured at the microscale. The phononic crystal slab of cross holes possesses a wide complete band gap extending from 1.5 to 4 MHz. The chain is formed by distributing resonators along an aperiodic path but respecting an equal coupling scheme. Collective modes of vibration of the chain are imaged thanks to a laser vibrometer. A fine spectrum of resonances involving the motion of all resonators at once is observed. Finally, a theoretical model is proposed to explain the frequency response and to predict the discrete spectrum of resonance frequencies.

II. METHODS

Recently, femtosecond laser-assisted wet etching has been demonstrated as a powerful mask-free micromachining method, allowing fabrication of various 3D microstructures in fused silica (a-SiO 2 ) and other glass materials with high spatial precision [START_REF] Bellouard | Fabrication of high-aspect ratio, micro-fluidic channels and tunnels using femtosecond laser pulses and chemical etching[END_REF] . This method is well suited for the fabrication of phononic crystals resonators due to the high selectivity of etching of glass materials in KOH (> 200), resulting in high aspect ratio structures with well-defined and smooth vertical sidewalls. In this work, the process parameters were adjusted to produce a significant increase in the etch rate of irradiated silica (> 200 times in KOH). Increased etch rate can be explained by laser-induced stress generation that is responsible for weakened chemical bonds (and hence bond angle variations) in the SiO 2 matrix 31 . Moreover, with increasing number of laser pulses, stress builds up and a porous structure is created that can be penetrated much faster by the etchant.

The chain of coupled resonators shown in Fig. 1 was manufactured in a fused silica plate. The sample size is 26 mm× 20 mm× 500 µm. A square lattice of cross holes, with lattice constant a = 714 µm, was etched in the plate by using a 3D laser system (FEMTOprint model f100 aHEAD Enhanced), equipped with a 5 W femtosecond laser operating at λ = 1030 nm. The overall fabrication process is a direct writing method (i.e., without mask) and is composed of three main steps. The process flow is shown in Figure 1(a). First, a laser machining process is programmed in 3D by use of a dedicated software (Alphacam) that results in the generation of a machining code file. Next, the fused silica substrate is scanned by the focused laser beam, according to the program, and the material is exposed to low-energy femtosecond pulses that accelerate locally the etching rate. Only a very thin contour (2-2.5 µm wide) of each cross-hole is exposed through the whole substrate thickness in sequential order. In the final step, the thin walls of exposed silica (aspect ratio ≈ 80) are double-side etched in a 10M KOH solution at 80 • C. The total etching time was 3h (etch rate ≈ 2.2 µm/min) resulting in releasing of silica cross blocks into the etchant. Ultrasonic vibrations were activated periodically during etching to enhance penetration of KOH through the high aspect structure and to stabilize the etching rate. The scanning electron microscope image of a cross hole (measured with SEM Quanta 450W) in Fig. 1(b) allows to verify the smoothness of the walls after etching. Fabrication errors are estimated to be less than ±1 µm.

Figure 1(c) shows a sketch of the experimental setup. The sample is held over a printed circuit board (PCB) by using screws and nuts. Its bottom surface is in contact with a piezoelectric ceramic transducer (PZT) that acts as a source of vibrations. We use a MEGGIT Pz27 (Navy II) soft relaxor type PZT whose fundamental resonant frequency is 5 MHz to cover the whole frequency range of interest. Pz27 is an all-round soft PZT material with good coupling factor, good charge coefficients, high Curie temperature, and low mechanical quality factor. We avoid resonance effects in the transducer by working below its fundamental resonance frequency. The transformed vertical displacement vibration signal is transmitted to the sample through a contacting screw.

We designed the chain of resonators by omitting the etching of selected holes, as shown in Fig. 1(d). We purposely chose the sequence of defects to follow a rather arbitrary path in the plane of the surface, with the intent of showing that the coupling of resonators allows to tunnel elastic waves without creating straight channels. Resonators are separated by (±2, ±1) or (±1, ±2) lattice shifts, i.e. their absolute separation distance is fixed by the direction of the chain jumps from resonator to resonator. To verify the precision of the fabrication, an optical microscope (Leica model DM8000) is used to check the dimensions of the fabricated structures as shown in Fig. 1(d). The resonators along the chain are numbered in sequence. The source region is placed inside the phononic crystal. For frequencies inside the complete phononic band gap this ensures that the source region is isolated from the frame surrounding the phononic crystal structure. Evanescent coupling decreases very fast with distance, resulting effectively in near-neighbor coupling along the chain. Since the first resonator is the only one close to the source region, it is the only resonator that is significantly coupled with it.

Measurements of vibrations of the chain are conducted by using a laser Doppler vibrometer (Polytec Microsystem Analyzer MSA-500, equipped with analog displacement decoder model DD-300; 0.03-24 MHz frequency response with 50 nm/V sensitivity). A periodic chirp is chosen as the source waveform in order to cover the frequency range of interest. The electric signal is transformed to a displacement vibration signal via the PZT transducer. The vertical, or out-of-plane, displacement signal is recorded by the vibrometer at the top surface of the sample.

All numerical simulations in this work are implemented using the finite element method in three dimensions (3D). The mechanical material parameters used for isotropic fused silica are mass density ρ = 2201 kg/m 3 , Poisson's ratio ν = 0.17, and Young's modulus E = 72.5 GPa. A 3D finite element model was set to compare with vibration experiments. The mesh encompasses the full 20 × 18 phononic crystal of cross holes, the chain of resonators, and the source. To minimize wave reflections at the lateral boundaries of the mesh, radiation boundary conditions are applied. To allow for a direct comparison with experiment, a time-harmonic and spatially random wave source of vertical polarization is applied at the source region [START_REF] Laude | Stochastic excitation method for calculating the resolvent band structure of periodic media and waveguides[END_REF] . With this setting, all possible modes of vibration are excited as the frequency is swept though the band gap range. The frequency response function (FRF) can be recorded at any position along the sample surface. Dispersion relations and phononic band structures are obtained by solving an eigenvalue problem with periodic boundary conditions applied at the external boundaries of the unit cell or super-cell. The phononic crystal slab has a complete band gap extending from 1.5 to 4 MHz. Material loss can be estimated with the Qf factor (product of quality factor by operating frequency in Hertz). Qf ≈ 5 10 12 for fused silica [START_REF] Laude | Phononic crystals: artificial crystals for sonic, acoustic, and elastic waves[END_REF] . With the largest frequency considered being 4 MHz, we then have in principle Q > 1.25 10 6 for the material loss limit. Given our experimental resolution, such fine line width can not be resolved in practice. As a result material damping was not added to numerical simulations. Note, however, that radiation damping, resulting from the finite number of phononic crystal rows around the resonator chain, is taken into account through the use of radiation boundary

conditions.

III. RESULTS

The experimental measurement of the frequency response function (FRF) is presented in Fig. 2(a). The line colors represent the FRF measured at different resonators. The FRF shows a series of sharp resonances within the complete phononic band gap, corresponding to vibration modes of the sample. We consider particularly in the following the frequency regions around 2.26 MHz, 2.53 MHz, and 3.55 MHz, where the clearest signals are observed. When looking closer, it is observed that each main response actually subdivides in a series of closely spaced sharp resonances, as shown in Fig. 2(c). The number of sub-peaks is of the order of 18 in each case, i.e. a number related to the number of coupled resonators in the chain. A similar observation was made previously regarding the channeled transmission spectrum of sonic crystal waveguides [START_REF] Wang | Channeled spectrum in the transmission of phononic crystal waveguides[END_REF] . The maximum amplitude of the vertical displacement of each resonance peak varies notably, indicating that vibration modes are variously matched to the excitation source. The vertical motion at the source location is shown in Fig. 2(e) for comparison. It can be observed that the resonance peaks measured at the location of the source do not match those measured inside the chain of resonators. This is expected, since at the location of the source thickness resonances [START_REF]Thickness resonances are standing Lamb waves of the homogeneous plate, having a zero lateral wavenumber[END_REF] of the fused silica plate should be excited and these have a different dispersion spectrum compared to the resonators. Interestingly, even when the source region vibrates very faintly, the vibrations of the chain of resonators can still be excited until the very last resonator.

Experimental results can be compared with the numerical results shown in Fig. 2(b) and(d). In the three frequency ranges extending around 2.26 MHz, 2.53 MHz, and 3.55 MHz, series of sharp resonances are obtained with a frequency spacing in fair agreement with experiment. Note that the experimental frequency resolution is limited to 1.56 kHz, due to a limitation on the number of sample points, while numerical computations are presented with a resolution of 0.2 kHz to ensure that each peak is resolved. Generally, numerical and experimental results agree fairly well, except for frequencies around 3 MHz, where a significant FRF is only observed in the numerical simulation. We attribute this small discrepancy to the actual experimental environment that differs from the numerical excitation source: the experimental excitation is not truly spatially random as is assumed in the numerical simulation. Furthermore, it may be that the excitated vibrations have too small amplitude to be observed over the experimental noise floor.

Figure 3 displays the displacement maps observed at selected frequencies in both experiment and simulation. Animations of the vertical displacement distribution at frequencies 2.2625 MHz, 2.5234 MHz, and 3.5578 MHz are further shown in Videos 1, 2, and 3. It is verified that all resonators oscillate collectively at the same frequency. In all cases, vibrations are mainly concentrated at the four corners of the resonators. A clear modal shape repeats for every resonator, in correspondence with the initial vibration mode of the resonator. The collective vibrations are for the whole chain of resonators, without apparent spatial attenuation along the chain, and are only limited by the experimental size of the array.

IV. MODEL OF THE PHONONIC POLYMER A. Discrete sequence of eigenfrequencies

The chain of coupled resonators forming a phononic polymer can be analyzed with various discrete models. Hamiltonian models have been proposed, especially in the context of topological phononics [START_REF] Wang | Topological phononic crystals with one-way elastic edge waves[END_REF][START_REF] Mousavi | Topologically protected elastic waves in phononic metamaterials[END_REF][START_REF] Süsstrunk | Observation of phononic helical edge states in a mechanical topological insulator[END_REF] . Following the Su-Schrieffer-Heeger (SSH) model of polyacetylene [START_REF] Su | Solitons in polyacetylene[END_REF] , a coupling Hamiltonian can be constructed for eigenstates combining the degrees of freedom describing each resonator. Solving for the Schrödinger equation with the coupling Hamiltonian gives the sequence of eigenfrequencies. Such a technique parallels the dynamical matrix approach used to describe the dispersion of phonons in atomic lattices. Here, we consider a classical version of such quantum techniques [START_REF] Maria Escalante | Dispersion relation of coupled-resonator acoustic waveguides formed by defect cavities in a phononic crystal[END_REF] where the state of the phononic polymer is described by one macroscopic degree of freedom per resonator, U n . All resonators are identical except for a spatial shift in the crystal and have the same isolated resonance frequency ω 0 . By isolated resonator, we mean a resonator embedded in the infinite crystal and placed very far from any other resonator. ten

The dynamical equation for coupled resonators is writ-

-Üm = N n=1 D(m, n)U n , (1)
with a symmetric dynamical matrix D(m, n) such that D(m, m) = ω 2 0 . The symmetry of the dynamical matrix is a consequence of reciprocity. Considering timeharmonic vibrations at a frequency ω, the dynamical equation leads to eigenfrequencies that are eigenvalues of the dynamical matrix:

ω 2 U m = N n=1 D(m, n)U n .
(

) 2
As a result, there are exactly N eigenstates of vibration of the phononic polymer. The symmetry of the dynamical matrix implies that the eigenfrequencies (ω i , i = 1 • • • N ) are real. This is the most general form we can obtain without making further simplifying assumptions.

If the resonators are distributed evenly along the chain, it can be assumed that the dynamical matrix is banded, i.e. D(m, n) = γ |m-n| , reflecting the idea of a translationally invariant chain. The coefficients γ |m-n| account for nearest-neighbor coupling. This particular assumption was made for coupled-resonator waveguides formed of an infinite chain of coupled resonators 40 . The dispersion relation for Bloch waves of the infinite periodic chain was then obtained as 39

ω 2 = 2 ∞ m=0 γ m cos(km∆). ( 3
)
We stress that this formula applies to the phononic polymer only in the limit N → ∞.

If we further assume that only nearest-neighbor cou-pling occurs, then the dynamical matrix simplifies to For this particular case, the eigenfrequencies can be obtained analytically and are

D =         
ω 2 N,m = ω 2 0 + 2γ cos πm N + 1 , m = 1 • • • N. ( 5
)
This result is derived in the Appendix, where it is shown that a sequence of orthogonal polynomials is formed, whose zeros are the eigenfrequencies. As a consequence, the eigenfrequencies of polymer chains with increasing number of resonators are interleaved. It is further obtained that they distribute on the continuous dispersion relation for the infinite polymer

ω 2 = ω 2 0 + 2γ cos(k∆) ( 6
)
where ∆ is the period of the chain and with the wavenumber being sampled according to k∆ = πm N +1 , m = 1 • • • N . This condition is equivalent to the phase-matching condition introduced to describe the channeled spectrum of the transmission of phononic crystal waveguides [START_REF] Wang | Channeled spectrum in the transmission of phononic crystal waveguides[END_REF] . As a generalization, we infer that if the dispersion relation ω 2 (k∆) is known, then the resonance frequencies will be given by ω 2 ( πm N +1 ), m = 1 • • • N . This distribution explains the sequence of sub-peaks in Fig. 2 around each initial defect resonance.

B. Comparison to coupled-resonator waveguide

Direct comparison of the discrete phononic polymer model with a coupled-resonator waveguide is not strictly possible since the former is aperiodic while the latter is periodic. Anyway, in order to enable comparison to some extent, we consider a simplified periodic version of the phononic polymer, as depicted by the super-cell in Fig. 4(a). Compared to the actual chain of coupled resonators considered in experiments, the coupled-resonator waveguide of Fig. 4(a) implements a sequence of lattice translations (2, -1) then (2, +1) so that the spatial period along axis x is 4a (four crosses along axis x in the super-cell). The coupled-resonator waveguide thus defined could be obtained by a continuous deformation of the chain of resonators.

Figure 4(b) shows the phononic band structure for the coupled-resonator waveguide whose super-cell is presented in Figure 4(a). The color bar measures the polarization amount of the vertical component of displacement. Compared to the phononic band structure of the perfect crystal, additional bands appear inside the complete band gap. Figure 4(c) shows close-up views of the band structure around 2.26 MHz, 2.52 MHz, and 3.555 MHz, corresponding to labels A, B, and C inserted in panel (a). As a note, the 4a period causes spurious foldings at the X and M points of the first Brillouin zone, since two periods of the chain are actually included in the super-cell. As a result, there are 2 bands in spectral ranges A and B, and presumably more in spectral range C. These bands have different polarization contents and thus couple differently with the source of vibrations. In particular, the mostly vertically polarized band in spectral range B has the cosine shape of Eq. ( 6), in correspondence with the single coupling coefficient of the dynamical matrix in Eq. ( 4). The other bands do not have this simple cosine shape and more coupling coefficients should be included to describe them. Significantly, all bands extend almost symmetrically toward the X and the M point, indicating that the coupling coefficients are here mostly independent of the direction of the coupling, as we have implicitly assumed in the discrete model. Eigenmodes at the high symmetry points of the Brillouin zone on three selected bands of spectral ranges A, B, and C are illustrated in Fig. 5. It can be seen that the eigenmodes in spectral range B are identical for all high symmetry points. We further verified that they are identical with the defect mode of the isolated resonator. More significantly, the modal shapes are clearly identical with those of Fig. 3, supporting the conclusion that the collective vibrations of the chain result from evanescent coupling of the individual resonators. In spectral range A, the situation is a bit more intricate, as two degenerate isolated defect modes co-exist. They form two separate bands when unfolded, with a dispersion involving more than one Fourier harmonic, i.e. corresponding to the general form (3) rather to the simpler (6). In spectral range C, the bands do not have a pure out-of-plane vibration character, and the discrete model of the phononic polymer should be enriched with up to three degrees of freedom per resonator, which we leave as a perspective.

As remarked in Section II, the band appearing around 3 MHz in the numerical simulation is not observed in the experiment. For completeness, however, we discuss briefly its characteristics. The eigenmodes at the M point, the Γ point and the X point of the first Brillouin zone are shown in Fig. 6(a). These eigenmodes are identical for all three high symmetry points. be mostly similar to those at 2.5 MHz. Finally, we compare in Appendix B the acoustic transmission rate between the phononic polymer and conventional phononic waveguides with a variable number of bends.

V. CONCLUSION

In this paper, coupled elastic vibrations of a chain of coupled resonators in a square lattice phononic crystal slab have been studied. Coupling of resonators is evanescent within the wide complete band gap obtained with cross holes, resulting in nearest-neighbor coupling. We designed the chain of resonators by omitting the etching of selected holes. Measurements show that collective vibrations of the evanescently coupled resonators are efficiently excited. We imaged the modal vibrations over the surface and observed that the resonators oscillate collectively at the same frequency. Significantly, collective oscillations survive although the chain is aperiodic and does not form a classical phononic crystal waveguide.

The number of resonances is as a rule equal to the number of resonators in the chain, in a first approximation. The evanescent coupling mechanism we consider is indeed reminescent of coupled-resonator waveguides. The main difference is that we do not require periodicity in the plane. This relaxed assumption allows us to design chains of resonators with a rather arbitrary shape.

This work provides a foreground for the consideration of phononic polymers, where resonators with equal free resonance frequency are arranged in an arbitrary chain and coupled evanescently. Active or even smart manipulation of localized resonators is thus expected. Such phononic polymers could be used for fusion sensing purposes, with each resonator functionalized to the same or different analytes. It is expected that the collective vibrations should be highly sensitive to a local change in one of the resonators. Phononic polymers could also be employed in optomechanics, using light forces such as radiation pressure to excite and detect collective resonances without any mechanical contact. Furthermore, by tuning the coupling coefficients between resonators, it should be possible to implement topological phononics models inspired for instance by the Su-Schrieffer-Heeger (SSH) model of the molecule of polyacetylene.

The fabrication process we employed relies on the locally-selective etching of fused silica. Other microfabrication techniques could obviously be alternatively selected, in view of different material platforms or to adjust the frequency range of the ultrasonic resonances. with the first two polynomials being P 0 = 1 and P 1 = -X. The next polynomials are P 2 = X 2 -1, P 3 = -X 3 + 2X, P 4 = X 4 -3X 2 + 1, and so on.

The existence of the recurrence relation (A2) implies that the set of the real polynomials P N is orthogonal. As a consequence, their roots are all real and are interleaved.

Figure 1 .

 1 Figure 1. Experimental setup for excitation and imaging of the vibrations of a chain of coupled resonators in a phononic crystal slab. (a) A array of cross holes forming a square lattice is etched in a fused silica plate by femtosecond laser-assisted wet etching. (b) A scanning electron microscope image of a single cross hole and a zoomed view at a sidewall, illustrating the fabrication roughness, are shown. (c) A sketch of the experimental setup shows the fused silica sample attached to a PCB by screws and nuts. The bottom surface is in contact with a piezoelectric patch (PZT) acting as a source of vibrations. Out-of-plane vibrations are detected at the top surface with a laser micro-system analyzer. (d) An optical microscope image of the sample shows the chain of 17 coupled resonators embedded in a 20 × 18 phononic crystal of cross holes and the position of the source of vibrations.

Figure 2 .

 2 Figure 2. (a) Experimental and (b) numerical absolute vertical displacements at the position of the resonators. Each line color is for a particular resonator as labeled in Fig. 1(b). The complete band gap extends in the white region. (c)-(d) Zoomed views around selected frequencies. (e) Absolute experimental vertical displacements at the location of the source of vibrations are shown for comparison.

Figure 3 .

 3 Figure 3. Vertical displacement maps at frequencies 2.2625 MHz, 2.5234 MHz, and 3.5578 MHz, in experiment (a) and numerical simulation (b), respectively. The color scale is for the amplitude of the vertical displacement from 0 (blue) to maximum (red). Numerical values are globally scaled to experimental values.

Video 1 .

 1 Animation of the experimental vertical displacement distribution at frequency 2.2625 MHz. Video 2. Animation of the experimental vertical displacement distribution at frequency 2.5234 MHz.

Video 3 .

 3 Animation of the experimental vertical displacement distribution at frequency 3.5578 MHz.

Figure 4 .

 4 Figure 4. Dispersion of an equivalent coupled-resonator elastic waveguide. Supercell (a), band structure (b), and zoomed band structures (c) around 2.26 MHz, 2.52 MHz, and 3.55 MHz, corresponding to point A, B and C, respectively. Blue (red) corresponds to zero (maximum) amplitude for the out-of-plane displacement field. The gray areas indicate the passing frequency ranges.

Figure 5 .

 5 Figure 5. Eigenmodes of the supercell shown at the M point, Γ point and X point of the first Brillouin zone at the passing bands corresponding to A, B, and C labeled in Fig. 4. Blue (red) corresponds to zero (maximum) amplitude for the displacement field.

Fig. 6 (Figure 6 .

 66 Figure 6. Characteristics of the band appearing around 3 MHz. (a) Eigenmodes are shown at the M point, the Γ point and the X point of the first Brillouin zone. (b) Numerical vertical displacement map at 2.9947 MHz for the chain of resonators. (c) Band structure of the equivalent coupledresonator elastic waveguide.

  VI. ACKNOWLEDGMENTSThis work was supported by the French RENATECH network and its FEMTO-ST technological facility. Financial support by the National Natural Science Foundation of China (11702017 and 11532001) is gratefully acknowledged. V.L. acknowledges financial support by the EIPHI Graduate School (ANR-17-EURE-0002). T.-T.W. acknowledges a scholarship provided by the China Scholarship Council (No. 201707090056).

Figure 7 .

 7 Figure 7. (a) Transmission spectra for three circuits with a different number of bends and for the aperiodic chain. The circuits are shown in (b).

  The recurrence relation actually resembles the one for Tchebychev polynomials, T N +2 -2XT N +1 + T N = 0. Tchebychev polynomials are characterized by the relation T N (cos θ) = cos(N θ) for 0 ≤ θ ≤ π, but are not the only set of orthogonal polynomials determined by this recurrence relation. In particular, their derivatives U N = 1 N T N also do. Those polynomials satisfy U N (cos θ) = sin(N θ)/ sin(θ) and are U 1 = 1, U 2 = 2X, U 3 = 4X 2 -1, U 4 = 8X 3 -4X and so on. It is easily checked that P N (X) = U N +1 (-2X). The zeros of U N +1 are given by condition sin((N + 1)θ)/ sin(θ) = 0, or θ m = mπ N +1 , m = 1 • • • N . As a result, the zeros of P N are λ m = -2 cos(θ m ) = -2 cos( mπ N +1 ), m = 1 • • • N and are all in interval [-2, 2].

Appendix B: Waveguide transmission

The sample we prepared was designed to observe the collective vibrations of the chain of resonators, but not specifically to demonstrate waveguiding. In order to obtain high transmission, it would be necessary to take care of the coupling of the source with the entrance of the waveguide, but also to reduce the reflection coefficient at the exit of the chain considered as a waveguide.

To compare the acoustic transmission rate between the phononic polymer and conventional phononic waveguides, we designed three additional and more traditional coupled-resonator waveguides with a different number of bends. Figure 7 shows the simulated transmission spectra. We calculated the transmission rate by

where U is the total displacement at the exit and U 0 is the total displacement at the entrance of the waveguide. The main observation that we make is that transmission spectra are similar for all waveguides and for the phononic polymer, though the details of the spectra (separation between resonances) depend on the total length of each chain. Compared to conventional coupledresonator phononic waveguides, a distinctive advantage of the phononic polymer is that guided waves can propagate along a rather arbitrary path and not only along principal directions of the crystal.


Isabelle Starkier

Errances d'hier et d'aujourd'hui :

Le Tango des étoiles errantes Introduction : migration, exil et errance Migration : action de passer d'un pays dans un autre en parlant d'un peuple (synonymes : émigration, immigration). Exil : expulsion hors de la patrie. Errant : Qui erre, qui n'est pas fixé. Qui voyage au hasard1 .

Trois définitions qui mettent en jeu le voyage, le déplacement mais avec cette différence que la migration et l'exil impliquent le regret, la séparation, le déchirement et que l'errance renvoie à l'impérieuse nécessité, subie ou agie, d'aller vers un futur toujours plus lointain, toujours en question, toujours en marchesans aucune allusion bien entendu. J'y vois là une différence de point de vue fondamentale dans la structuration des récits ou des spectacles qui vont en être faits.

La migration, l'immigration et l'exil nous entrainent dans leurs voyages qui ont tous un commencement et une fin, hélas souvent tragiques. Ils vont donc prendre la forme d'un récit : monologue, témoignages ou récit épique comme a pu le faire magistralement Ariane Mnouchkine avec son Dernier Caravansérail2 . Nous sommes dans la narration linéaire qui porte et déporte des êtres humains d'un bord à un autre du monde pour tenter de (re) construire l'identité dont ils sont exilés en endossantavec douleur le plus souvent, avec espoir parfoisune nouvelle identité culturelle. Au mieux ils s'intègreront en désintégrant partie ou totalité de leur ancienne culture, religion, langue à l'exemple de l'immigration des années 50 en France. Au pire, ils s'identifieront à leur ancienne identité, considérée comme perdue ou fantasmée en désintégrant la nouvelle identité qui leur est assignée, rêvant d'un impossible retour au point de départet ce à travers deux ou trois générations comme en témoigne ce mal-être de générations entre deux eaux, celle des enfants de l'immigration aujourd'hui. Récit d'un continent perdu, nostalgique parfois, haï souvent pour les violences qu'on y a subi; récit d'une arrivée et du chantier où il faut construire une nouvelle parole, une nouvelle culture. L'exil contribue à cette épopée d'une patrie perdue ou rejetée. Dans tous les cas nous sommes face à une tentative de mise en paroles et en images par un récit fondateur de l'histoire de migration entre deux pays (ou plus) qui s'ancre le plus souvent dans le présent du passage, écartelé entre ces deux pôles.

Dans la vague du « théâtre-récit » qui émerge un peu partout en Europe, le théâtre de l'immigration algérienne s'affirme comme un théâtre du « dire » et de la « parole », dans une logique de témoignage et de narration d'une expérience, à la fois individuelle et collective. Ce théâtre use de l'emprunt et mêle ainsi étroitement la structure dramatique de type occidental avec les normes de la tradition orale algérienne. Du processus de création des spectacles à l'insertion de multiples instances narratives frontales, le théâtre de l'immigration algérienne opère une mise en récit du quotidien répondant à une exigence de témoignage3 C'est ainsi que Jeanne la Gallic parle du mouvement théâtral qui naît dans les années 70 autour de l'immigration algérienne notamment. On est dans des récits de vie qui oscillent entre un ici et un là bas : « Cette double culture, va et vient constant entre le pays d'origine et le pays d'accueil est l'une des caractéristiques du répertoire du théâtre de l'immigration algérienne, en particulier avec la troupe Nedjma »4 .On songe aux oeuvres de Kateb Yacine, à Baba la France de Rachid Akbal etc…Mais on peut également penser aujourd'hui encore aux fresques épiques telles qu'Ariane Mnouchkine les porte avec Le dernier Caravansérail, ou plus récemment encore avec Simon Abkarian et son Envol des cigognes5 , grandiose «saga » mythologique, qui laisse entendre la migration que lui-même, en tant qu'Arménien libanais, a vécu. Nous ne sommes plus dans un récit de vie au quotidien qui primait durant les années 70 où le choc de l'immigration et de l'exil dévorait toute autre parole plus poétique ou plus métaphorique mais dans l'épopée qui rattrape cet ailleurs et la rejoint ici et maintenant pour mieux l'écrire, l'inscrire et l'intégrer, faute de l'être… Aujourd'hui, la violence du voyage des migrants occupe le devant du terrain et engloutit, si je puis dire, les deux pôles d'un passé enfui et d'un futur incertain. Le présent tragique des migrants offre, avec la tendance d'un théâtre documentaire, une matière théâtrale que Sylvie Roques développe dans son article. Le témoignage, celui des migrants comme celui des enfants d'immigrés, est le texte sur lequel se construit l'histoire racontée « en miroir ».

L'errance comme parole

Or l'errance refuse le récit car elle n'est pas linéaire. Elle n'a pas de but : le voyage en constitue la continuité sans fin. Il n'y a ni intégration, ni désintégration mais « assimilation » qui est une traversée d'autrui, comme le rappelle sans cesse Emmanuel Levinas 6 . C'est un voyage en forme de questionnement, qui dès qu'il se pose sait qu'il repartira vers de nouveaux voyages, vers de nouvelles questions. C'est un voyage qui s'enrichit de ce qu'il est voyage, à l'image de Don Quichottedont on interprète l'oeuvre comme cabalistique. Il n'a pas de but car il est lui-même son propre but. C'est donc à ce voyage là d'un peuple sans terre qui ne peut qu'errer, passant à travers l'immigration comme une étape et jamais comme un ancrage, traversant autrui sans jamais s'y fondre malgré ses tentatives désespérées d'y parvenir que j'ai souhaité convier le public autour du spectacle que j'ai mis en scène et dont le titre se veut évocateur : Le Tango des étoiles errantes.7 L'errance diasporique devient intrinsèque à la culture, à la philosophie du peuple juif Vivre dans la diaspora du sens, n'être jamais « chez soi », nier jusqu'à la possibilité même de se voir institué en sujet souverain, la dispersion de toute unité de référence, selon Lévinas, enrichit d'autant plus celui qui l'exerce qu'une sensibilité à l' altérité devient la vertu de cette existence hors frontière. Toutefois, la beauté de l'exil, c'est qu'il porte la charge d'une aspiration: « La terre promise ne sera jamais dans la Bible une « propriété », au sens romain du terme, et le paysan, à l'heure des prémices, ne pensera pas aux liens éternels qui le rattachent au terroir, mais à l'enfant d'Aram, son ancêtre qui fut un errant » (Lévinas, 1957d, p. 33).À l'opposé du modèle d'Ulysse, la réflexion ne saurait être première. Difficile liberté ne dessine pas une configuration du judaïsme par la figure du cercle, du reste impensable, mais par un amoncellement de lignes et de traces auxquelles la fixation d'un centre échappe8

Le théâtre juif procède de l'interdit de la représentation et d'une éthique iconoclaste qui déchiffre le monde comme un Livreet traduit les images en mots. De l'interprétation critique et signifiante du monde naît l'image -et non l'inverse, qui devient alors la représentation d'une analyse (dans tous les sens du terme). La voix suscite le voir comme il est dit dans l'Exode « Tout le peuple voit les voix »9 .

Cette esthétique de la lettre, non comme véhicule d'images mais comme sonorité conceptuelle, « s'infigure », en se disant elle-même sans rien dire d'autre : comme un autocommentaire de sa propre énonciation qui s'agirait elle-même, de l'intérieur. On se parle, comme Dieu se parle à lui-même « A la venue de Moshe dans la tente du rendez-vous, pour lui parler, il entend la voix lui parler…Il lui parle »10 . Et Philon, reprenant le fait que le peuple d'Israël « voit la voix » de Dieu, expliquera à son tour que la voix de Dieu est visible car « les mots que prononce la voix de Dieu/envoyée par Dieu sont si clairs qui ils semblent être vus plutôt qu'entendus (De Decalogo, 46) 11 .Il offre alors également une autre interprétation du verset « Tout le peuple vit la voix » (Exode 20, 18) selon laquelle la voix de dieu est visible parce que toutes les paroles que Dieu prononce sont des actes et non des mots (De Decalogo, 47) »12 C'est le commandement premier du Naasse ve Nichma (Agis puis Pense) qui peut sembler obscurantiste à un premier degré (agir sans penser) mais qui fait d'une parole l'acte même et de la voix un personnage. Il n'y a pas de Dieu, il y a un Verbe. D'après ce principe fondateur, le théâtre juif est un théâtre de l'énonciation et non de l'énoncé. Il conserve ce rapport à la parole agie, agissante, créative (au Verbe donc) et non à la parole reçue (au texte ou récit mythiquebiblique).

C'est par cette parole que Le Tango des étoiles errantes s'ouvre, parole qui est traitée comme un personnage : « Moi, je suis le tango ». De là ce personnage ou ce non-personnage du tango traverse les paroles qu'il suscite, les paroles de l'errance, qui le définissent en même temps qu'il les définit. Pour en décrire la dramaturgie, partant de cette rencontre du tango et du klezmer, j'ai articulé les trois grandes périodes des migrations du peuple juif : l'Europe de l'Est où le tango yiddish naît, s'épanouit puis traverse les pogroms jusqu'à Auschwitz ; l'Amérique de la comédie musicale où il émigre ; et enfin l'Argentine où il accompagne la souffrance de la traite des juives mais aussi l'exil des nombreux immigrants d'Europe de l'Est. C'est plus à un jeu de pistes qu'à un parcours que le spectateur est convié, qui va de paroles en paroles, de musiques en musiques, chacune créant sa propre contrée imaginaire.

Judit -L'immigrant, c'est un passé qui ne revient pas.

Alain-Et l'exilé, c'est un déracinement qui se répète toujours.

Judit -Le grand-père trancha au couteau la corde qui l'avait lié/ A ces paysages agrestes de chênes et de vergers/ Mais dans son âme se sont accrochés/Les morceaux d'une Europe qu'il ne peut oublier Alain -Le petit-fils trancha au couteau la corde qui l'avait lié / A cette ville de suie et de tango blessée/ Mais dans son âme se sont accrochés/ Les rues de Buenos-Aires qu'il ne peut oublier. Car c'est la parole, qui est acte, qui fait acte et qui convoque l'errance. Ce n'est pas un récit qui représente, qui met en images ou en mots, ce sont les mots qui font et défont l'histoire du Peuple du Livre. C'est ce davar hébraïque qui signifie à la fois mot et chose, parole et évènement qui s'épanouit dans le midbar à l'étymologie semblable, c'est-à-dire dans la traversée du désert, l'acte fondateur de l'errance du peuple juif.

Judit

Tout récit engendre potentiellement une mythologie. Le récit de la migration est particulièrement en quête de son fatum. Celui qui le porte d'un point à un autre, qui donne du sens à un voyage qui paradoxalement rejoint une forme d'absurdité tragique. Le récit de la migration est révolte. La révolte implique la quête de sens, qui conduit à une lutte contre le fatum. Lorsqu'Emmanuel Levinas désigne la « tentation d'idolâtrie » à travers « l'immanence ou athéisme foncier du voir et du savoir » 14 , il renvoie au processus idolâtre de constitution du sens qui bloque l'interprétation au profit de la représentation. La représentation devient un mode de pensée où la pensée s'idôlatre elle-même dans l'objet offert et qui la représente : une pensée « pensant la chose » 15 A l'opposé, la parole qui ne suscite pas d'image ou de représentation, suivant en cela le deuxième commandement du « Tu ne feras point d'image.. », demeure un signe de la pensée, un signe qui, selon Jean-Joseph Goux, dans Iconoclastes, « n'est pas un symbole mais quelque chose de tout à fait indifférent et sans valeur, sur quoi l'esprit exerce un pouvoir de détermination » 16 .

Peut-on dire alors que la parole de l'errance constitue en soi et par soi-même une pensée de l'errance ? Dans Le Tango des étoiles errantes, le personnage du tango accompagne ce concept intrinsèque d'un voyage sans but ni fin qui par définition est errance et traverse les continents et l'Histoire en constituant le juif errant, celui dont l'identité même tient à l'errance.

En ce sens il est dans l'impossible représentation : il reste et demeure parole, il n'a de personnage que par son absence de personnage ou l'incarnation de son propre concept. Il est en creux : sa présence s'inscrit dans son absence même -sa fuite, son exil. C'est un dibbouk 17 qui s'exprime sans corps, qui n'est que parole agissante, actante, actrice. Pour tracer comme on suit à la trace cette errance musicale du peuple juif au vingtième siècle, j'ai d'ailleurs cherché d'autres mots à associer à ce non-récit : des mots d'Albert Cohen, de Paul Celan, de Kafka, d'Albert Londres. Chaque fragment de texte renvoie par diffraction ces mots à d'autres mots afin de faire signe et non de faire sens. Ils constituent la musique verbale de notre musique de tango. Il évite ainsi toute tentation d'illustration où l'on prend l'image pour la chose, le mot pour l'objet. 18 Si on revient aux principes de la mimesis, fondement de la représentation théâtrale, le récit tourne insensiblement à l'allégorie. Chaque image est moins le véhicule d'une expression signifiante que la réactualisation incantatoire d'un évènement sacralisé dans un temps circulaire qui est celui de la représentation (à l'image du choeur antique !) et que définit Maître Eckhart quand il parle « d'un anneau circulaire dont le centre est partout et la circonférence nulle part » 19 (p.84). Le théâtre comme microcosme de ce macrocosme qu'est le monde, la petite histoire témoignant pour la grande : le signe fait sens par ce monde auquel il renvoie; le récit prend tout son sens par ce monde dans lequel il s'ancre.

Pour se référer encore un fois au travail de Jeanne la Gallic, le théâtre de l'immigration algérienne est Théâtre du verbe et de la parole.. (qui) favorise le témoignage direct et le récit intime dans lesquels la scène devient une sorte de confessionnal ou d'exutoire. Ces récits de vie permettent au narrateur de faire le point sur son existence, témoignant d'une situation sociale ou individuelle susceptible de concerner le plus grand nombre. Les situations narrées représentent ce qui est vécu, individuellement et collectivement 20 Elle évoque à travers une citation d'Alicia Lindon un théâtre « dans lequel ces expériences narrées acquièrent un sens, en provoquant la connexion des événements quotidiens et des situations particulières vécues ou imaginées » 21 . Le témoignage devient universel et porte alors le récit en images des migrations pour tenter de dire la migration en général, le déchirement des identités. Ainsi le récit de l'immigration permet la dimension allégorique et épique puisque Il se produit alors une traduction de la dimension intime et personnelle des expériences, en formes socialement partagées. Le récit est donc extirpé de la privatisation et rendu visible dans l'espace public collectif. Le narrateur se place dans une posture de témoignage, de porte-parole d'un vécu susceptible de concerner le plus grand nombre 22

Le tango comme traversée d'autrui

(Judit, chuchoté) -

Envoie-moi Une lumière pour mon chemin Du soleil pour mes larmes Qui coulent depuis si longtemps Qu'on n'en verra jamais la fin

Elle chante : Shik mir a shtral… (Alain et Judit, paroles tuilées en boucle) -Un matin, au sortir d'un rêve agité, Gregor Samsa s'éveilla transformé en une épouvantable vermine…"Que m'est il arrivé?" songea-t-il. Ce n'était pas un rêve… La musique les interrompt avec Le tango d'Auschwitz 19 Cité par Ernst CASSIRER, La philosophie des formes symboliques, Tome 2 (La pensée mythique), Editions de Minuit, Paris 1972, p.30. 20 Jeanne LE GALLIC, Ibid. 21 LINDON Alicia, « autobiographique, reconstruction de l'expérience et fabulation : une approximation à l'action sociale », in Sociétés, n° 87, janvier 2005, p. 57 22 LE GALLIC, Jeanne, Ibid.

L'errance tout au contraire traverse et brise la circularité de ce monde de correspondance puisqu'elle est sans fin. C'est une traversée sans but qui n'a d'autre temps que le présent, et pour notre Tango des étoiles errantes, d'autre temps que la musique. La musique détourne la parole en concept pur, en grammaire intérieure comme la qualifiait Kandinsky : « Plus dégagé est l'élément abstrait de la forme, et plus le son en est pur, élémentaire » [START_REF] Vassily | Du spirituel dans l'art et dans la peinture en particulier[END_REF] . Contre le récit universel de la connaissance (récit fondateur du poète qui prétend dire ce que sont les choses) s'élève la musique de ce tango de l'errance du peuple du Livre et c'est par la musique de ce tango yiddish, de cette traversée par la culture yiddish et/ou klezmer du tango que j'ai abordé l'histoire des « migrations » du peuple juif au vingtième siècle [START_REF]La comédienne apparait au micro, le visage éclairé[END_REF] .

« Le son peut donc tout aussi bien être une lettre de l'alphabet, car ici, le visible et l'audible sont également réduits à n'être que de simples signes de l'esprit » 25 . Ce rapport du son à la lettre renvoie à la tradition musicale juive d'un art qui ne devient pensée que lorsque le signifiant disparaît et avec lui le rapport dialectique du sujet et de l'objet pour laisser place à la seule conscience du « soi central » : un soi qui se parle à lui-même sous forme de concepts et en arrive, comme la petite sonate de Vinteuil dans La recherche du Temps perdu, à ne rien dire d'autre que lui-même.

Le langage esthétique est alors un langage conceptuel il n'est plus le possédé mais la représentation de ce qui le possède. Il exprime cette intériorité sans objet ou ce « son de l'élément abstrait » 26 dont parle Kandinsky ou encore une « subjectivité abstraite qui est un moi entièrement vide et sans aucun contenu » [START_REF] Wilhem | [END_REF] Comme le suggère Hegel, il n'y pas extériorisation …le besoin d'art en général a donc ceci de rationnel que l'homme, en tant que conscience, s'extériorise, se dédouble, s'offre à sa propre contemplation et à celle des autres. Par l'oeuvre d'art, l'homme qui en est l'auteur cherche à exprimer la conscience qu'il a de luimême 28 Il y aurait plutôt un phénomène d'intériorisation : au contraire d'un monde des correspondances qu'est le monde des images -symboles et allégories -, qui travaille sur l'extériorisation de l'objet projeté, représenté, l'errance renvoie à ce concept pur ou à cette subjectivité abstraite de l'intériorisation identitaire. C'est encore une fois le « Je suis qui je suis » de Don Quichotte -que j'ai monté il y a quelques années-arrivé au bout de sa route, d'un chemin sans fin dont on a d'ailleurs noté qu'il était cabalistique.

La musique porte à sa limite extrême le paradoxe d'une figuration de l'infigurable -qu'on le nomme divin ou métaphysique. Il fait du non-dit un dit, du signe un son et concrétise l'abstraction à travers l'immatérialité de ses mélodies : la musique…art représentatif (pour) une chose qui de sa nature ne peut jamais faire l'objet d'une représentation (…) la musique (est) la copie d'un modèle qui lui-même ne peut jamais être représenté directement29 La petite sonate de Vinteuil chez Marcel Proust est la clef d'une « recherche du Temps Perdu » qui joue sur l'errance du narrateur et de son impossible narration d'un temps fixe, figé : passé, présent, futur…Il n'y a pas de nostalgie, pas de retour en arrière à proprement parler mais un voyage à travers les tempsqui en ce sens est abstraction musicale, petite phrase d'une sonate perdue. C'est à ces temps-là que se réfère le Tango des étoiles errantes. Le tango yiddish nous interpelle dans le rapport du son à la lettre : les paroles du tango, paroles de souffrance et d'humour s'il en est, ne racontent guère qu'en surface l'histoire, la petite histoire de ces juifs en Pologne ou en Argentine en miroir de la grande Histoire qui les porte et qu'ils font ou défont au gré de leurs errances. Ils sont plutôt poésie pure, c'est-à-dire comme la musique avec laquelle ils sont étroitement mêlés, abstraction intérieure d'un concept : celui d'une errance identitaire, au fondement de l'identité d'un peuple de la Question et de « sa traversée d'autrui ». Le tango yiddish détourne le tango qu'il « s'assimile » : il en fait un chemin, le chemin d'une quête à travers des continents psychiques plus que géographiques.

Le chant du peuple sans terre

La question qui est donc posée à travers la mise en représentation des migrations ou des errances des peuples déracinés, bannis, exilés, déchirés entre plusieurs pays est aujourd'hui nodale. C'est en effet à ce noeud-là de la représentation bien plus que du vécu, du témoignage, de l'insupportable présent de ces êtres brisés, meurtris, noyés, détruits, que s'accroche l'impossible quête d'identité au coeur de la question de l'immigrationqui englobe migrants et errants. Une nouvelle génération d'êtres dont on n'a pas réussi l'intégration parce que précisément cette notion-là est à remettre en question au profit de l'assimilation. Elle se raccroche au regret d'un paradis perdu qui serait une destination finale et devient souvent mortifère. Lorsqu'on se réfère à la note d'intention d'Ahmed Madani avec son magnifique spectacle Illuminations 30 dont il dit qu'il se veut une « fable historique » :

Le processus passe par un retour en arrière, une sorte de remontée dans l'histoire des migrations en France. Partir de l'intime et du vécu de chacun afin de dégager les grandes lignes permettant de décrypter la situation des banlieues aujourd'hui conduit à l'émergence d'une figure symbolique qui se définit par la quête identitaire : « Je ne suis pas un Algérien, je suis un immigré 31 Bien que les spectacles d'Ahmed Madani, en témoignant de la perte de repères de générations que l'on a privé d'une mémoire postcoloniale entachée de déni et de mauvaise conscience, témoignent (au sens propre du terme puisqu'ils partent de témoignages) d'une réelle intégration et d'un merveilleux optimisme, ils se construisent sur ce même récit de migration qui tend à l'épopée. Il vient travailler ce discours théâtral sur les migrants, fait de témoignages, « d'instants de vie » rassemblés pour en dire la continuité et le sens.

La représentation de l'errance, que ce soit à travers notre spectacle ou les pièces du théâtre yiddish (celles de Sholeim Aleikhem 32 , d'Itzik Manger 33 , de Peretz 34 ou de tant d'autres…) contribuent à la traversée des chemins dont le voyage seul constitue la musique. On assiste au décryptage d'une identité derrière laquelle on court sans cesse et qui signe donc l'errance comme un acte fondateur et sa représentation comme les éclats diffractés d'un miroir sans tain, d'un texte sans fin.

  -Dans un bateau plein de rêves d'immigrés/ Son espérance cherchait un port où s'ancrer/ Ou pouvoir planter son identité/Pour trouver la force de recommencer/ Sachant que le passé ne revient jamais Alain -Dans un bateau plein de rêves d'exilés/ Son espérance cherchait un port où s'ancrer/ Ou pouvoir planter son identité/ Pour trouver la force de recommencer/ Sachant que le passé peut bien se répéter Judit -Le grand-père pense au ciel où scintillaient les étoiles de son enfance/ Et il s'endort/ Bercé par le son d'un bandonéon Alain -Le petit fils pense au ciel où scintillaient les étoiles de son enfance/ Et il aimerait s'endormir / Bercé par le son d'un bandonéon 13

14

  Emmanuel LEVINAS, « Interdit de la représentation et « droits de l'homme »,in L'interdit de la représentation, Colloque de Montpellier, 1981, Editions du Seuil, Paris, p.108 15 Idem, p.109 16 Jean-Joseph GOUX, Les Iconoclastes, Ed du Seuil, Paris, 1978, p.79 17 Le thème folklorique du dibbouk nomme, dans la tradition juive cabalistique, un esprit entrant dans le corps d'un vivant, à la suite d'une erreur ou d'une mauvaise action. C'est l'histoire d'une possession et elle a donné naissance à une pièce fondatrice du Théâtre de la Habima (premier théâtre en hébreu qui est devenu le Théâtre National en Israël), pièce de Shalom An-Ski écrite et créée initialement en yiddish en 1917 à Vilna, (Le Dibbouk ou Entre deux mondes) puis par Vsevolod Meyerhold à Moscou en 1922. 18 (Alain lit un article d'Albert Londres) -Ils sont cinq mille. Par groupe de vingt-cinq à trente, ils envahissent la ville. Les Juifs trouvés dans les tramways sont jetés à terre, le tram en marche. Tout passant, même celui qui n' a ni l'habit ni la barbe, mais un peu d'Israël au milieu du visage est rossé. Ils visitent les cafés, les restaurants, et vident à coups de botte les consommateurs non chrétiens. Des équipes, armés de marteaux et de gourdins, défoncent les vitrines des magasins juifs. Enfin la ruée vers les synagogues. Tout est brisé à coups de hache. Ils s'emparent des livres saints et des "fiancées couronnées". Ils les déchirent, les marquent mettent le feu et dansent autour de l'incendie en bénissant les flammes….

Définition du Littré

Le Dernier Caravansérail, Création collective sous la direction d'Ariane MNOUCHKINE, Théâtre du Soleil (2003), filmée par Ariane MNOUCHKINE, Arte Editions, 2006.

Jeanne LE GALLIC, « Le théâtre de l'immigration algérienne des années 1970 : un théâtre du « dire » » in Corinne ALEXANDRE-GARNER et Isabelle KELLER6PRIVAT (dir.), Collection Sciences Humaines et Sociales, Migrations, exils, errances et écritures [en ligne], Presses universitaires de Paris Nanterre, Nanterre, 2012, p.101-114. Disponible sur Internet : <http://books.openedition.org/pupo/2068>. ISBN : 9782821850842. DOI : https://doi.org/10.4000/books.pupo.2068.

Jeanne LE GALLIC, Le théâtre de l'immigration algérienne dans les années 1970: un espace de représentation et de témoignage, in Synergies Algérie n°10 -2010, p.160. https://gerflint.fr/Base/Algerie10/le-gallic.pdf

L'envol des cigognes (une tragi-comédie de quartier), écrit et mis en scène par SimonABKARIAN, 2017.

« L'homme juif, déclare Levinas, découvre l'homme avant de découvrir les paysages et les villes. Il est chez soi dans une société, avant de l'être dans une maison. Il comprend le monde à partir d'autrui, plutôt que l'ensemble de l'être à partir de la terre. » Ami BOUGANIM, La rime et le rite. Essai sur le prêche philosophique, L'Harmattan,Paris, 1996, p. 40.

https://vimeo.com/233634431

Gilbert LAROCHELLE, « Liberté et justice chez Lévinas. L'expérience de l'impossible », in Revue Philosophique de Louvain., Quatrième série, tome 102, n°4, 2004. p.592. https://www.persee.fr/doc/phlou_0035-3841_2004_num_102_4_7576

La Bible traduite parAndré CHOURAQUI, Exode, 20, 20., Desclée de Brouwer, 1989

Idem, Nombres 7,89

Laurence WUIDAR. "Voix visible et musique intelligible chez Philon d'Alexandrie." Divus Thomas, vol. 119, no.1, Edizioni Studio Domenicano,,Bologne, 2016, p.175. JSTOR, www.jstor.org/stable/48503808

Ibidem

La Vueltas de la Vida, tango écrit par Judit MAIAN/Martina INIGUEZ, musique: José Ojivieki. Le petit diaogue qui précède fait partie du texte du spectacle…La traduction est de nous.

Georg Wilhem Friedrich HEGEL, Op.cit., Tome 1, p.62

Arthur SCHOPENHAUER, Le monde comme volonté et comme représentation,Editions PUF, Paris, 1966, p.328


Manon Ruffin

Jeanne Bigot

Claire Calmel

Julia Mercier

Andrés Pizzorno

Manuel Rosa-Calatrava

Harriet Corvol

Viviane Balloy

Olivier Terrier

Loïc Guillot
email: loic.guillot@inserm.fr

Flagellin from Pseudomonas aeruginosa modulates SARS-CoV-2 infectivity in CF airway epithelial cells by increasing TMPRSS2 expression

Keywords: SARS-CoV-2, lung, epithelium, TMPRSS2, Cystic Fibrosis, Pseudomonas aeruginosa, flagellin

The major challenge of the COVID-19 health crisis is to identify the factors of susceptibility to SARS-Cov2 in order to adapt the recommendations to the populations and to reduce the risk of getting COVID-19 to the most vulnerable people especially those having chronic respiratory diseases including cystic fibrosis (CF). Airway epithelial cells (AEC) are playing a critical role in the immune response and in COVID-19 severity. SARS-CoV-2 infects the airways through ACE2 receptor and the host protease TMPRSS2 was shown to play a major role in SARS-CoV-2 infectivity. Here, we show that the main component of P. aeruginosa flagella, ie. flagellin is able to increase TMPRSS2 expression in AEC, and even more in those deficient for CFTR. Importantly, this increased TMPRSS2 expression is associated with an increase in the level of SARS-CoV-2 infection. Considering the urgency of the health situation, this result is of major significance for patients with CF which are frequently infected and colonized by P. aeruginosa during the course of the disease.

As of November 3, 2020, coronavirus disease 2019 (COVID-19) pandemic caused by the Severe Acute Respiratory Syndrome (SARS)-Coronavirus (CoV)-2 has infected globally near 47 million people with more than 1 million deaths (https://covid19.who.int). One of the major challenges of this health crisis is to identify factors of susceptibility to this viral infection in order to adapt public health recommendations and to reduce the risk of getting COVID-19, notably for the most vulnerable people. This is the case of patients having common chronic respiratory diseases (CRD) such as asthma and chronic obstructive pulmonary disease (COPD), but also for patients with less common or rare CRD, such as cystic fibrosis (CF) and interstitial lung diseases. Even if it seems reasonable to think that, as they have a lung impairment, people with CRD would be at higher risk to develop a severe COVID-19, the magnitude of this risk is still uncertain [START_REF] To | A Global Respiratory Perspective on the COVID-19 Pandemic: Commentary and Action Proposals[END_REF]. Along with the clinical follow-up of these patients to have a better estimation of their risk, basic research regarding the pathophysiology of SARS-CoV-2 infection will provide important insight. This is particularly true for patients with CF. This disease is caused by variants in the CFTR gene, the most frequent being F508del, which result in an abnormal function of the airway epithelial cells (AEC). During the course of CF, lungs of the patients are inflamed and chronically infected by various pathogens including Pseudomonas aeruginosa, which is the most prevalent (80% in CF adults) [START_REF] Elborn | Cystic fibrosis[END_REF]. Up to now, a multinational report identified 40 cases of CF patients infected by SARS-CoV-2 with no death recorded (3). Of these cases, 50% had chronic P. aeruginosa airways infection, 78% had symptoms at COVID-19 onset such as fever (77% of the symptomatic patients), altered cough (48%) and/or dyspnea (32%). AEC are known to play a critical role in both the immune response and COVID-19 severity (4). SARS-CoV-2 infects the airways mainly through the ACE2 receptor, and two specific host proteases, TMPRSS2 and FURIN, were shown to play a major role in SARS-CoV-2 infectivity [START_REF] Coutard | The spike glycoprotein of the new coronavirus 2019-nCoV contains a furin-like cleavage site absent in CoV of the same clade[END_REF][START_REF] Hoffmann | SARS-CoV-2 Cell Entry Depends on ACE2 and TMPRSS2 and Is Blocked by a Clinically Proven Protease Inhibitor[END_REF][START_REF] Walls | Structure, Function, and Antigenicity of the SARS-CoV-2 Spike Glycoprotein[END_REF][START_REF] Yan | Structural basis for the recognition of the SARS-CoV-2 by full-length human ACE2[END_REF][START_REF] Zhou | A pneumonia outbreak associated with a new coronavirus of probable bat origin[END_REF].

In this study, we show that the main component of P. aeruginosa flagella, ie. flagellin (PA-F), is able to increase TMPRSS2 expression in airway epithelial cells, and even more in those deficient for CFTR. Importantly, this increased TMPRSS2 expression is associated with an increase in the level of SARS-CoV-2 infection.

We first examined ACE2, FURIN and TMPRSS2 expression from a previous transcriptomic study performed with primary human airway epithelial cells (hAEC) isolated from control donors (non-CF) and CF patients homozygous for the CFTR F508del variant, infected by P. aeruginosa [START_REF] Balloy | Normal and Cystic Fibrosis Human Bronchial Epithelial Cells Infected with Pseudomonas aeruginosa Exhibit Distinct Gene Activation Patterns[END_REF]. At baseline (time 0 h), similar ACE2 and FURIN mRNA expression levels are observed between non-CF and CF primary hAEC (Fig. 1A, left), whereas TMPRSS2 expression is significantly increased in CF primary hAEC (Fig. 1A,right). Importantly, P. aeruginosa increases TMPRSS2 mRNA expression over time (Fig. 1A) in CF primary hAEC only, while ACE2 and FURIN expressions are not affected (Fig. 1A,left).

We thus exposed CF primary hAEC to flagellin, the ligand of Toll-Like Receptor (TLR) 5 and the most important proinflammatory factor of P. aeruginosa present in the sputum of CF patients [START_REF] Balloy | Flagellin concentrations in expectorations from cystic fibrosis patients[END_REF]. We observed that flagellin significantly increases mRNA levels of TMPRSS2 without increasing those of ACE2 (Fig 1B).

In order to study the mechanism underlying this increase, we used the Calu-3 cell line, which exhibits high ACE2 and TMPRSS2 mRNA levels and detectable ACE2 protein expression compared with Beas-2B and 16HBE airway epithelial cell lines (Fig. 1C). This result is consistent with the already observed higher ability of SARS-CoV-2 to replicate in Calu-3 cells in comparison to Beas-2B (6).

Next, as observed in CF primary hAEC, we showed that exposure of Calu-3 cells to PA-F significantly increases TMPRSS2 transcripts, without affecting ACE2 mRNA (Fig. 1D) and protein expression (not illustrated). Intriguingly, the increase in TMPRSS2 expression is more important in Calu-3 cells deficient for CFTR (Calu-3-CFTR-KD) than in standard Calu-3 cells sufficient for CFTR (Calu-3-CFTR-WT) (Fig. 1D). This increase in TMPRSS2 expression is TLR5 dependent (Fig. 1E) and also detected at the protein level (Fig. 1F). As expected, flagellin induces the synthesis of the proinflammatory cytokine IL-8 (Fig. 1G) both in CFTR-WT and -KD Calu-3 cells. As already described [START_REF] Blohmke | Innate immunity mediated by TLR5 as a novel antiinflammatory target for cystic fibrosis lung disease[END_REF], CF-like epithelial cells exhibit a more potent inflammatory response to flagellin.

Finally, after infection with SARS-CoV-2, we observed an increase in the intracellular viral mRNA nsp14, which is significantly higher in Calu-3-CFTR-KD cells and even more important when cells were pre-stimulated with flagellin (Fig. 1H,left). In contrast, extracellular viral mRNA nsp14 level measured at the apical side as a surrogate of viral production is significantly lower in Calu-3-CFTR-KD than in Calu-3-CFTR-WT cells (Fig. 1H,right). Whereas flagellin pre-stimulation does not affect viral particle release in Calu-3-CFTR-WT cells, a lower, yet not statistically significant, level of nsp14 mRNA is measured at the apical side of whereas, Calu-3-CFTR-KD cells.

The increase of intracellular viral mRNA found in CF cells exposed to flagellin, indicating a higher level of infection, is likely the result of the increased expression of TMPRSS2. Indeed, TMPRSS2 inhibition by the serine protease inhibitor camostat mesylate is sufficient to prevent infection with SARS-CoV-2 [START_REF] Hoffmann | SARS-CoV-2 Cell Entry Depends on ACE2 and TMPRSS2 and Is Blocked by a Clinically Proven Protease Inhibitor[END_REF]. We could hypothesize that the lower level of viral particle at the apical side observed in CF cells may be the result of increased host defense capacities of CF cells against viral infection. Further mechanistic works are necessary to know if this response would be protective or damaging for CF patients. Most of CF patients are colonized by P. aeruginosa and, to date, even if the results of the clinical follow-up of CF patients are limited, it seems that these patients are not at higher risk to severe COVID-19 than the general population (3). Several studies have delineated the antiviral capacities of flagellin against other respiratory virus, including influenza A (13), and flagellin was recently suggested to modulate the innate response in order to eliminate SARS-CoV-2 and resolve COVID-19 [START_REF] Golonka | Harnessing innate immunity to eliminate SARS-CoV-2 and ameliorate COVID-19 disease[END_REF]. Whether CF patients are protected of developing severe form of COVID-19 thanks to their P. aeruginosa infection will be difficult to prove at the clinical level. However, further works elucidating the mechanisms associated with the specific host response of CF cells to SARS-CoV-2 infection could help to elucidate this question and give future therapeutic insights. Chanson, University of Geneva, Switzerland) cells were cultured in MEM-Glutamax (ThermoFisher scientific) supplemented by 10% FCS (Eurobio, Les Ulis, France), 1% nonessential amino acids (ThermoFisher scientific), 10 mM HEPES (ThermoFisher scientific), 1% sodium pyruvate (ThermoFisher scientific) and 1% antibiotics (ThermoFisher scientific) as previously described [START_REF] Pizzorno | Characterization and Treatment of SARS-CoV-2 in Nasal and Bronchial Human Airway Epithelia[END_REF]. Primary CF hAEC (from a male Caucasian donor aged 32 with CF) were cultured as recommended by the manufacturer using hAEC complete culture media (Epithelix, Geneva, Switzerland). Beas-2B cells were cultured in F12 media supplemented by 10% FCS, 10 mM HEPES, and 1% antibiotics. 16HBE14o-were generously given by Dieter Gruenert (originator) and Dr. Beate Illek (provider) from Unversity of California San Francisco (UCSF). They were cultured in MEM-Glutamax supplemented by 10% FCS, and 1% antibiotics as recommended by the provider.

RT qPCR

RNA was isolated using a NucleoSpin RNA/miRNA (Macherey Nagel, Duren, Germany). RT was performed using a high-capacity cDNA kit (Applied Biosystems, Foster City, CA, USA).

Real-time qPCR was performed with an ABI QS3, using Sensifast Probe Lo-Rox Kit (Biotechnofix, Guibeville, France), TaqMan probes and cDNA as a template. For relative quantification, the amount of target genes was normalized to the expression of GAPDH relative to reference group (specified in the figure legends) used as a calibrator and was calculated using the 2 -ΔΔCt method.

Western Blotting

Total proteins were extracted using RIPA buffer (Euromedex, Souffelweyersheim, France).

An equal amount of proteins (20 μg) was reduced, size-separated on 12% stain-free precast SDS-polyacrylamide gels (Biorad, Hercules, CA), and transferred to nitrocellulose membranes using iBlot2 apparatus (ThermoFisher scientific). The membranes were blocked in 5% milk in TBS-Tween 0.1% and incubated with specific primary antibodies O/N at 4°C.

Bound antibodies were detected using clarity chemiluminescent substrate (Biorad). Images were recorded with a Fujifilm LAS-3000 bioimaging system (Fujifilm, Stamford, CT, USA).

Immunofluorescence

Cells were plated in 24-well plate on 12 mm diameter #1.5 coverslips (Marienfeld, Lauda-Königshofen, Germany) or on filters at the air-liquid interface and growth as previously described. After treatments, cells were rinsed with PBS and fixed with ice cold PFA 4% 20 min. The cells were permeabilized 10 min with 0.1% Triton X-100 in PBS; and then, washed with PBS and incubated in saturation solution (PBS+BSA 5%) for 1h. Cells were then incubated overnight at 4 °C with primary antibodies in PBS supplemented with 1% BSA. The following day, the cells were washed 3 × 5 min with PBS and incubated for 1h at room temperature with secondary antibodies, followed by DAPI staining. Coverslips were sealed with ProLong diamond mounting media (ThermoFisher scientific). Fluorescence microscopy was achieved using a Olympus fluorescent microscope.

ELISA

Concentrations of human IL-8 (R&D, Minneapolis, MN, USA) were measured in cell supernatants using ELISA kit according to the manufacturer's instructions. The 3,3′,5,5′tetramethylbenzidine substrate was from Cell signaling technology.

Statistical analysis

Differences among groups were assessed for statistical significance using Prism

Figure. 1 .Calu- 3 (

 13 Figure. 1. (A) (left) Heatmap of ACE2, TMPRSS2 and FURIN expression (fold change) and

  Figure. 1

Acknowledgements

LG received a grant from the Faculté de Médecine Sorbonne Université (AAP COVID19).

We acknowledge Pr. Marc Chanson (University of Geneva) for generously providing Calu-3-CFTR-WT and Calu-3-CFTR-KD cells (15). We acknowledge Pr. Dieter Gruenert and Dr.

Beate Illek (University of California San Francisco (UCSF) for generously providing 16HBE14o-.

Author Contributions

LG, VB, MR and JB designed experiments, MR, JB, CC, VB, OT, AP, JM and LG made the experiments. LG wrote the manuscript. MR, JB, VB, MRC and HC revised the manuscript.

Conflict of Interest

Authors have no conflict of interest to declare.

Ethics

This project is approved (Opinion number 20-688) by the Inserm Institutional Review Board (IRB00003888, IORG0003254, FWA00005831).

Figure Legend

Supplementary information

Materials and Methods

Reagent