Julie Meimoun

Choltirosn Sutapin

Grégory Stoclet

Audrey Favrelle

Pascal Roussel

Marc Bria

Suwabun Chirachanchai

Fanny Bonnet
email: fanny.bonnet@univ-lille.fr

Philippe Zinck
email: philippe.zinck@univ-lille.fr

Lactide Lactone Chain Shuttling Copolymerization Mediated by an Aminobisphenolate Supported Aluminum Complex and Al(OiPr) 3 : Access to New Polylactide Based Block Copolymers

Keywords: Chain shuttling copolymerization, Polylactide, Aluminum complex, Block copolymers, Polyesters

The chain shuttling ring-opening copolymerization of L-lactide with -caprolactone has been achieved using two aluminum catalysts presenting different selectivities and benzyl alcohol as chain transfer agent. A newly synthesized aminobisphenolate supported aluminum complex affords the synthesis of lactone rich poly(L-lactide-co-lactone) statistical copolymeric blocks, while Al(OiPr)3 produces semi-crystalline poly(L-lactide) rich blocks. Transalkoxylation is shown to operate efficiently. The crystalline ratio and glass transition temperatures of these new class of polylactide based block copolymers can be tuned by adjusting the catalysts and the comonomers ratio.

The depletion of petroleum resources and the advent of plastic pollution are growing human concerns. The development of biobased and biodegradable/compostable polymers can help in this regard. Among the different biobased materials available on the market, polylactide is becoming more and more competitive and its production is continuously growing. This compostable, biocompatible and bioresorbable polyester is synthesized by ring-opening polymerization (ROP) of lactide, a cyclic ester obtained from oligomerization-cyclization of lactic acid. L-lactide (Figure 2) is particularly interesting since it leads to a 100 % isotactic poly(L-lactide) or PLLA, a semi-crystalline material which displays mechanical properties close to those of conventional polyethylene terephthalate (PET). The brittleness of PLLA and its low elongation at break (2-6%) hampers however its development for numerous applications. [START_REF] Liu | Research progress in toughening modification of poly(lactic acid)[END_REF] In order to overcome these problems and to allow PLLA to be used for a wider range of applications, it is commonly used in the form of blends with other polymers 1 or in composites. [START_REF] Murariu | PLA composites: From production to properties[END_REF] Another strategy consists in copolymerizing L-Lactide with other cyclic esters displaying complementary properties which will enhance the thermomechanical properties of the final material. A challenge is thus to synthesize advanced high quality polylactide or related structures by dedicated synthesis platforms. In this respect catalytic polymerization is relevant, and more specifically chain shuttling copolymerization (CSP, represented Scheme 1), a powerful tool allowing the access to multiblock / segmented copolymers with original microstructures in a one-pot one-step route. 3 Scheme 1. Chain shuttling copolymerization (CSP). M are metals with their ligands L. CTA is the chain transfer agent.

In this process initially developed for ethylene / 1-alkene systems, [START_REF] Arriola | Catalytic production of olefin block copolymers via chain shuttling polymerization[END_REF][START_REF] Wei | Programmable Modulation of Co-monomer Relative Reactivities for Living Coordination Polymerization through Reversible Chain Transfer between "Tight" and "Loose" Ion Pairs[END_REF][START_REF] Descour | Exploration of the effect of 2,6-(t-Bu)2-4-Me-C6H2OH (BHT) in chain shuttling polymerization[END_REF][START_REF] Vittoria | Molecular Kinetic Study of "Chain Shuttling" Olefin Copolymerization[END_REF] the growing macromolecular chain is allowed to "shuttle" by transmetallation via an organometallic chain transfer agent (CTA) between two catalysts presenting a difference in comonomers reactivity ratios, and thus leading to statistical copolymeric blocks of different composition (Scheme 1). The block rich in ethylene is semicrystalline, while the higher amount of 1-alkene hampers the crystallization of the second type of block, giving rise to materials with valuable properties. It is worth mentioning that 1600 polymerizations experiments were realized by high-throughput technology in order to find the adequate catalytic combination based on bis(phenoxyimine) Zr and pyridylamide Hf affording the seminal shuttling. [START_REF] Arriola | Catalytic production of olefin block copolymers via chain shuttling polymerization[END_REF] If chain shuttling homopolymerization systems, where only one monomer is involved, have been well explored, 7- 14 it appears that since the seminal work in 2006 on ethylene/1-alkene comonomers, [START_REF] Arriola | Catalytic production of olefin block copolymers via chain shuttling polymerization[END_REF] the synthesis of original multiblock microstructures by chain shuttling copolymerization was extended to only one other comonomers combination, i.e styrene / conjugated diene, by rare earth based catalysts. [START_REF] Pan | Chain-Shuttling Polymerization at Two Different Scandium Si-tes: Regioand Stereospecific "One-Pot" Block Copoly-merization of Styrene, Isoprene, and Butadiene[END_REF][START_REF] Valente | Isoprene-Styrene Chain Shuttling Copolymerization Mediated by a Lanthanide Half-Sandwich Complex and a Lanthanidocene: Straightfor-ward Access to a New Type of Thermoplastic Elasto-mers[END_REF] This is probably to be ascribed to the complexity of the catalytic system required and the high level of constraints as soon as two different monomers are involved. We present herein the first example of a ring-opening chain shuttling copolymerization of L-lactide with the seven-membered -caprolactone, yield new original PLLA based block copolymers. Chain shuttling copolymerization is based on Coordinative Chain Transfer Polymerization, or CCTP, a degenerative chain transfer process. [START_REF] Kempe | How to Polymerize Ethylene in a Highly Controlled Fashion?[END_REF][START_REF] Valente | Coordinative Chain Transfer Polymerization[END_REF] In CCTP, the growing macromolecular chain transmetallates from the catalyst metal to the metal of the CTA, often a main group metal, via transalkylation, allowing the growth of several chains per catalyst atom (Scheme 2, bottom part). This can be combined with a second catalyst that displaces the growing chain to yield e.g linear α-olefins. [START_REF] Valente | Coordinative Chain Transfer Polymerization[END_REF] In the present work involving cyclic esters as comonomers, the chemistry is based on the socalled immortal ROP concept introduced by Inoue et al. using an aluminium porphyrin/alcohol system and an epoxide as monomer (Scheme 2, top). [START_REF] Asano | Immortal' polymerization. Polymerization of epoxide catalysed by an aluminium porphyrin-alcohol system[END_REF] Immortal ROP presents conceptual similarities with CCTP. The catalyst gives rise to the growth of several polymer chains by reversible transalkoxylation reactions. Immortal ROP was later transposed to cyclic esters, [START_REF] Endo | Immortal polymeriza-tion of. epsilon.-caprolactone initiated by aluminum porphyrin in the presence of alcohol[END_REF] including also other metal complexes (see for example ref. [START_REF] Miola-Delaite | Anionic coordinated polymerization of ε-caprolactone with aluminium, zirconium and some rare earths alkoxides as initiators in the presence of alcohols[END_REF] ) and their statistical copolymerization. [START_REF] Li | Copolymerization of ε-Caprolactone and l -Lactide Catalyzed by Multinuclear Aluminum Complexes: An Immortal Approach[END_REF]

Scheme 2. Similarity between Coordinative Chain Transfer Polymerization (bottom) and Immortal Ring-Opening Polymerization (top)

In order to introduce PLLA blocks in the targeted copolymer microstructure, a catalyst displaying a very high selectivity toward lactide has to be introduced in the reactive medium, while the second one will have to statistically copolymerize the two monomers with a substantial incorporation of the lactone. Such different behaviours can be reached by playing on the ligand framework. The metal catalysed statistical copolymerization of lactides with lactones is still quite challenging due to the fact that oxophilic metals involved in this reaction preferentially coordinate the lactide monomer. [START_REF] Duda | Block and random copolymers of єcaprolactone[END_REF][START_REF] Stirling | Catalytic metal-based systems for controlled statistical copolymerisation of lactide with a lactone[END_REF] A bulkier ligand is thought to rebalance the reactivity ratios of both comonomers and enables a better incorporation of the typically less reactive lactone. [START_REF] Stirling | Catalytic metal-based systems for controlled statistical copolymerisation of lactide with a lactone[END_REF] Bulky ligand should thus be considered for a high lactone incorporation, while less hindered catalysts should allow a preferential insertion of the lactide monomer. Both catalysts must also enable reversible transalkoxylation reactions, i.e the immortal-like character We have synthetized a new aminobisphenolate-supported aluminium complex by treatment of N,N-bis(2-hydroxy-3,5-dimethylbenzyl)cyclohexylamine (H2O2N CH ) with an equimolar amount of AlEt3 in toluene at room temperature. The ORTEP structure is presented in Figure 1, while synthetic and crystallographic details are given in the SI section. This complex, referred to as AlEtO2N CH affords, in combination with benzyl alcohol, the statistical copolymerization of L-lactide with -caprolactone with a high lactone insertion, as shown entry 1 in Table 1. Furthermore, in the presence of several equivalents alcohol CTA (entry 2), the good agreement between experimental and calculated number-average molecular weight (Mn) highlights an immortal ring-opening statistical copolymerization of Llactide with -caprolactone. Narrow molecular weight distribution can be noticed in both cases, with a dispersity of 1.11 for entry 2. Regarding the second catalyst required for the chain shuttling copolymerization system, Al(OiPr)3 has been reported in the literature for its ability to allow immortal-like ROP of both lactide and -caprolactone, together with a very high selectivity for the lactide in the course of statistical copolymerizations. [START_REF] Jacobs | Macromolecular Engineering of Polylactones and Polylactides. 5. Synthesis and Characterization of Diblock Copolymers Based on Poly-c-caprolactone and Poly(L,L or D,L)lactide by Aluminum Alkoxides[END_REF] These behaviours are also observed in our experimental conditions, as shown entry 3 in Table 1. Chain shuttling copolymerization experiments performed with the combination of these two catalysts, represented Figure 2, top part, are reported Table 1, entries 4-7 with increasing quantities of the chain transfer agent. Again, controlled molecular weights can be noticed, with dispersities decreasing down to 1.11 with increasing quantity of CTA, together with a very good agreement between calculated and measured number-average molecular weight. This attests to an efficient transalkoxylation / shuttling reaction occurring during the polymerization.

Figure 2. Chain shuttling copolymerization (top) and DOSY analyses of entry 7 (bottom)

A Diffusion Ordered SpectroscopY (DOSY) analysis was performed on a low molecular weight sample (low molecular weight is recommended for DOSY) realized with 9 eq. CTA (entry 7). The measurement of the diffusion coefficient allows to check if lactide and lactone based repeating units are on the same macromolecule or if it is a mixture of PLLA and the statistical copolymer that would have been made separately by the two catalysts. Indeed, in the former case, they will have the same diffusion coefficient, whereas in the latter case, two diffusion coefficients will be detected. The DOSY analysis given in Figure 2 bottom part shows that the lactyl -CH signal, the lactyl -CH3 and all the caprolactoyl -CH2 have the same diffusion coefficient, confirming the occurrence of chain shuttling. The splitting of the lactyl -CH signal around 5.1 ppm and that of the caprolactoyl -O-CH2 signal around 4.1 ppm is attributed to two different enchainments, i.e lactyl / lactyl and lactyl / caprolactoyl for the former, and caprolactoyl / caprolactoyl and caprolactoyl / lactyl for the latter. This can be quantified, and more information about the microstructure is given in the SI section. The thermal analysis of the so formed copolymers shows two glass transition temperatures and one melting temperature. Wide Angle X-ray Scattering (WAXS) analyses given in the SI have confirmed that the latter corresponds to crystalline PLLA. Tg1 is ascribed to the soft poly(L-lactide-co- caprolactone) blocks made by the aminobisphenolate complex, while Tg2 corresponds to amorphous poly(L-lactide-co--caprolactone) segments rich in lactide produced by Al(OiPr)3.

The thermal properties of the blocks show that the transalkoxylation rate is lower than the insertion rate in these conditions, allowing a multiblock structure with blocks of length sufficient to reach, e.g, the crystallisation of PLLA and the presence of 2 different Tg. When higher amounts of CTA are introduced, the dispersity goes down, suggesting that, even if lower than the insertion rate, the transfer rate is probably important. From 13 C Nuclear Magnetic Resonance (NMR) analysis we have evidenced that transesterification is not occurring in the course of the polymerization (see SI section for more details). The ratios of the catalysts and the comonomers were varied in order to assess which extent of microstructure can be reached with this original chain shuttling copolymerization system. An experiment with a lower amount of Al(OiPr)3 is reported entry 8. As expected, the lactide conversion is lower, 62% vs. 83% in entry 4. This leads also to a lower conversion of the lactone, probably due to the fact that the concentration of lactide in the reactive medium is higher, and thus it may be more competitive for the statistical copolymerization on the second catalyst. The Tg of the soft block increases as a consequence. Decreasing the amount of AlEtO2N CH does in turn not affect substantially the amount of lactide in the copolymer, as seen in entry 9, but does decrease the lactone conversion, as expected. As a consequence, the glass transition temperature of the soft block increases by 15 °C. The most important feature of this experiment is the important poly(L-lactide) crystalline ratio reached, with a melting enthalpy of 18.1 J/g, corresponding to a PLLA crystalline ratio close to 30%. c can be increased up to 40% by increasing the amount of lactide in the reactive medium, as presented entry 10. The Tg of the PLLA rich phase decreases down to 42 °C. The high crystalline ratio of the sample leads probably to a lower amount of lactide in the amorphous phase, which may explain the lower Tg. The increase of the quantity of lactone in the reactive medium leads in turn to the disappearance of the crystalline phase (entry 11), and thus to a fully amorphous block copolymer. Higher molecular weight can also be reached with this system by increasing the monomer over catalyst ratio, as shown entries 12-13. The immortal-like character of the polymerization is still observed for this ratio, as, in the presence of excess benzyl alcohol, the calculated and theoretical molecular weights are still in agreement together with a dispersity of 1.18 (entry 14). In this set of experiments, the propagation rate decreases as a consequence of an increased monomer / catalyst ratio, while the transfer rate probably stays constant, which may lead to a decrease of the size of the blocks. In addition, in entry 14 the amount of CTA is increased by a factor 4, which is a second leverage to decrease the size of the blocks, which may explain the disappearance of the crystallinity of the polymer. We thus have a fully amorphous block copolymer with PLA and statistical copolymeric blocks.

The nanostructure of the resulting block copolymers was then assessed via Small Angle X-ray Scattering. The results obtained for entry 6 showed the presence of an organized structure at the nanometer scale (see SI). This nanostructure consists of crystalline PLLA lamellae of thickness around ~5.6nm regularly stacked with a repeating distance of ~23nm as determined from the modeling of the SAXS curves. We finally assessed the mechanical properties of the new materials. We selected for this purpose samples with the highest crystallinity (entries 9 and 10), a lower crystallinity with a high molecular weight (entry 12) and finally a fully amorphous sample (entry 14). The results are reported in Table 2. All samples revealed a rather high elongation at break, from 50-75% for semi-crystalline samples to ca. 300% for the fully amorphous multiblock. The former exhibited in turn the highest modulus and strain at break, highlighting the key role of crystallinity. Toughness of 1.2-1.76 MJ/m 3 were obtained for semicrystalline materials. The data also underline the variety of mechanical behavior that can be obtained and the possibility to easily tune them with this new system. In summary, we report herein for the first time a lactide lactone chain shuttling copolymerization at two different aluminum sites, allowing to access new PLLA based block copolymers, where one of the block consists of a statistical, amorphous copolymer where the Tg can be varied, while the other block consists of semi-crystalline or amorphous poly(L-lactide) based material. Work is under progress to extend the scope of application of this new synthetic approach, and to characterize thoroughly the structure and properties of the resulting materials.

Figure 1 .

 1 Figure 1. Displacement ellipsoid plot of AlEtO2N CH (50%). Hydrogen atoms were hidden for clarity. Selected bong lengths (Å) and angles (°): Al-O1 = 1.7510(10); Al-O2 = 1.7408(9); Al-N = 2.0230(10); Al-C1 = 1.9515(13); O1-Al-O2 = 114.74(5); O1-Al-C1 = 110.50(5); O2-Al-C1 = 110.75(5)We have synthetized a new aminobisphenolate-supported aluminium complex by treatment of N,N-bis(2-hydroxy-3,5-dimethylbenzyl)cyclohexylamine (H2O2N CH ) with an equimolar amount of AlEt3 in toluene at room temperature. The ORTEP structure is presented in Figure1, while synthetic and crystallographic details are given in the SI section. This complex, referred to as AlEtO2N CH affords, in combination with benzyl alcohol, the statistical copolymerization of L-lactide with -caprolactone with a high lactone insertion, as shown entry 1 in Table1. Furthermore, in the presence of several equivalents alcohol CTA (entry 2), the good agreement between experimental and calculated number-average molecular weight (Mn) highlights an immortal ring-opening statistical copolymerization of Llactide with -caprolactone. Narrow molecular weight distribution can be noticed in both cases, with a dispersity of 1.11 for entry 2. Regarding the second catalyst required for the chain shuttling copolymerization system, Al(OiPr)3 has been reported in the literature for its ability to allow immortal-like ROP of both lactide and -caprolactone, together with a very high selectivity for the lactide in the course of statistical copolymerizations.[START_REF] Jacobs | Macromolecular Engineering of Polylactones and Polylactides. 5. Synthesis and Characterization of Diblock Copolymers Based on Poly-c-caprolactone and Poly(L,L or D,L)lactide by Aluminum Alkoxides[END_REF] These behaviours are also observed in our experimental conditions, as shown entry 3 in Table1. Chain shuttling copolymerization experiments performed with the combination of these

Table 1 . L-Lactide/-caprolactone (LA/CL) statistical and chain shuttling copolymerizations using AlEtO2N CH and Al(OiPr)3 at 70 °C in toluene, 1 M, 24 h.

 1

	En-	LA/CL/AlEtO2N CH /	Conv.	Yield	Mn calc	Mn exp	D c	LA/CL	Tg1	Tg2	Tm	H	c
	try	Al(OiPr)3/BnOH	LA/CL(%) a	(%)	(g/mol) b	(g/mol) c		(%) d	(°C) e	(°C) e	(°C) e	(J/g) e	(%) f
	1	250/250/1/0/1	83/61	56	47 300	41 400	1.22 54/46	-6	-	-	-	-
	2	250/250/1/0/5	79/59	53	9 100	9 000	1.11 54/46	-5	-	-	-	-
	3	250/250/0/1/1	86/4	45	8 000	9 700	1.19	94/6	-	44	144	32	38.5
	4	500/500/1/1/1	83/40	58	20 600	20 100	1.25 68/32	1	48	74/102	2.8	6.2
	5	500/500/1/1/2	86/45	65	17 500	16 200	1.22 67/33	0	47	74/103	1.3	2.9
	6	500/500/1/1/4	76/43	57	11 300	11 300	1.15 66/34	3	48	87	1.6	3.7
	7	500/500/1/1/9	79/28	44	6 100	7 100	1.11 71/29	-1	48	102	2.3	4.5
	8	500/500/1/0.6/1	62/29	38	21 900	23 100	1.16 64/36	7	-	110	1.4	3.3
	9	500/500/0.6/1/0.6	77/28	46	19 900	21 400	1.26 78/22	16	47	102	18.1	29.5
	10	700/300/0.6/1/0.6	70/20	43	21 500	24 600	1.27 88/12	6	42	144	30.1	40.7
	11	300/700/0.6/1/0.6	77/30	43	15 900	17 300	1.21 55/45	-8	57	-	-	-
	12 g 1000/1000/0.6/1/0.6	77/35	54	41 900	43 300	1.34 69/31	2	47	99	3.3	6.7
	13 g	1000/1000/1/1/1	77/36	52	38 000	37 900	1.30 62/38	-1	52	111	2.3	6.1
	14 g	1000/1000/1/1/4	78/52	62	24 500	23 300	1.18 66/34	4	56	-	-	-

a Conversion determined by

[START_REF] Liu | Research progress in toughening modification of poly(lactic acid)[END_REF]

H NMR analysis (see SI). b Mn calc = [LA / (3 x Al(OiPr)3 + BnOH)] x 144.14 x conv. LA + [CL / (3 x Al(OiPr)3 + BnOH)] x 114.14 x conv. CL. c Determined by Size Exclusion Chromatography. Mn corrected by a factor of [0.58 x wt%LA units + 0.56 x wt%CL unit]. d Molar composition of the isolated polymer determined by 1 H NMR (see SI). e Determined by Differential Scanning Calorimetry. f Estimated PLLA crystalline ratio (see SI). g 2M, 48h.

Table 2 . Mechanical properties of the copolymers meas- ured by tensile tests

 2

	Entry	E a	MAX b	B c	Toughness
		(MPa)	(MPa)	(%)	(MJ/m 3 )
	9	63.2	3.5	57	1.76
	10	63.6	1.7	74	1.08
	12	24.6	2.0	75	1.20
	14	0.8	0.2	275	0.15

a Young modulus. b Yield stress. c Elongation at break.

AUTHOR INFORMATION Corresponding Authors

* Fanny Bonnet (fanny.bonnet@univ-lille.fr) and Philippe Zinck (philipe.zinck@univ-lille.fr)

Funding Sources

The authors gratefully acknowledge the Agence Nationale de la Recherche (ANR) for partial funding through the PLANAVETTE project (ANR-21-CE06-0024). This work was also partially funded by the FWV ALPO Interreg Grant and the authors thanks the European Regional Development Fund (FEDER) and the University of Lille. Chevreul Institute (FR 2638), Ministère de l'Enseignement Supérieur de la Recherche et de l'Innovation, Région Hauts de France and CNRS are also acknowledged for supporting and funding partially this work. The French Ministry of Foreign Affairs and the French Embassy in Bangkok is gratefully acknowledged for the French-Thaï PHC SIAM project.

ACKNOWLEDGMENT

This paper is dedicated to the memory of Prof. André Mortreux, one of the fathers of CCTP, Coordinative Chain Transfer Polymerization, who passed away last year. We also thank Aurélie Malfait and Dr. Jonathan Potier for SEC measurements, Céline Delabre for Elemental Analysis, and Dr. Yohan Champouret and Prof. Marc Visseaux for fruitful discussion on the topic of L-lactide / lactone statistical ring-opening copolymerizations.


Romain Bellet
email: bellet@lma.cnrs-mrs.fr

Bruno Cochelin

Philippe Herzog

Pierre-Olivier Mattei

Experimental study of energy pumping in acoustics

This paper presents an experimental vibro-acoustic set-up that aims to reproduce the energy pumping phenomenon between an acoustic medium and an essentially nonlinear oscillator. It shows a one-way irreversible transfer of energy between the first acoustic mode in a tube and a thin visco-elastic membrane. Various observed aspects of energy pumping are presented : behaviors under sinusoidal forcing, pumping threshold, resonance capture and transient response.

Introduction

This paper deals with the energy pumping phenomenon for acoustic applications. The concept of energy pumping has been introduced by Vakakis and Gendelman [START_REF] Gendelman | Energy pumping in nonlinear mechanical oscillators, Part I : Dynamics of the underlying Hamiltonian systems[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators, Part II : Resonance capture[END_REF][START_REF] Vakakis | Inducing passive nonlinear energy sinks in vibrating systems[END_REF] in 2001 : the idea is to passively reduce the vibrations in a linear system (discrete or continuous) by attaching to it an essentially non linear damped oscillator. Under suitable conditions, a one-way irreversible transfer of energy can occur from the linear system to the nonlinear one, leading to an efficient cancellation of the oscillations in the linear system.

Many recent papers have been published around this interesting new concept, and to its application to various fields. For instance, the idea of using several nonlinear oscillators to improve the pumping has been investigated in [START_REF] Gourdon | Energy pumping with various nonlinear structures : numerical evidences[END_REF], and a first experimental validation of pumping has been demonstrated in [START_REF] Mc Farland | Experimental study of nonlinear energy pumping occuring at a single fast frequency[END_REF] on a two degrees of freedom mechanical system.

In this paper we succintly describe a simple two degrees of freedom model of which the main goal is to design an experimental set-up that permits to observe the energy pumping between an acoustic medium and a thin visco-elastic membrane. Then we present our experimental results. The first is the observation of different regimes under harmonic forced excitation between which a pumping threshold appears, corresponding to a non linear modes bifurcation. The resonance capture phenomenon [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators, Part II : Resonance capture[END_REF] is also highlighted. The other results are the noise level reduction in permanent regime and the faster sound extinction in transcient response resulting from energy pumping. This first verification of energy pumping in acoustics is a first step toward the design of new generation of passive acoustic absorbers that would be efficient at lower frequencies, where absorber materials are ineffective.

Models and design of the experimental set-up

The energy pumping phenomenon consists to damp a linear system by adding a non linear absorber and the simplest way to observe it is to consider a two degrees of freedom system (figure 1) consisting of two oscillators connected by mean of a small stiffness coupling spring. The first oscillator is a classical linear spring-mass system. The second one is composed of a mass, a nonlinear cubic spring (without linear stiffness), and a damper. Rescaling time with the natural period of the linear oscillator, the non dimensional equations for the positions u 1 (t) and u 2 (t) of the masses are :

ü1 + u 1 + β(u 1 -u 2 ) = 0
(1)

γ ü2 + α 3 u 3 2 + µ u2 + β(u 2 -u 1 ) = 0
with γ the mass ratio of the two masses, β the small coupling coefficient, µ the damping factor in the nonlinear oscillator and α 3 the cubic stiffness coefficient. The free solutions of (1) have been already discussed in [START_REF] Cochelin | Experimental evidence of energy pumping in acoustics[END_REF] to illustrate the energy pumping. The periodic solutions of the undamped system have also been analysed to show the connection between the pumping and the non linear modes of the system.

Our aim is to experimentaly reproduce the pumping phenomenon in a system where the linear oscillator is an acoustic medium. So we replace the linear mechanical oscillator by the air vibrating on the first acoustic mode in a tube, the coupling stiffness by the air in a coupling box and the nonlinear oscillator by a thin visco-elastic membrane as presented on figure 2. A continuous model associated to this principle can be obtained by taking Helmholtz equation for the air contained in the tube, and the nonlinear plate equation of the Von Karman type for the thin membrane. A Kelvin-Voigt model is adopted to account for the viscosity in the membrane : the Piola-Kirchhoff second stress tensor S is related to the Green-Lagrange strain tensor E and strain velocity tensor Ė by S = D : (E + η Ė), where D is a classical fourth order Hooke elastic tensor and η is the damping factor. As a first approximation, the continuous model can be reduced to a two degrees of freedom system by taking u(x, t) = u(t)cos( πx L ) for the air in the tube (first undamped mode), and w(r, t) = q(t)(1 -( r R ) 2 ) for the transversal displacement of the circular membrane (parabolic shape function). The two coordinates u(t) and q(t) correspond to the displacement at the end of the tube and at the center of the membrane. In the box, the acoustic velocity is supposed to be negligible and the pressure is thus considered as spatially uniform. It is related to the volume variation by p = ρc 2 ∆V

V with ∆V = u(t)S t -q(t) hSm 2 . Applying a classical Galerkin method, we get the following reduced system :

ü + u + β(u -q) = 0 (2) γ q + α 3 (2ηωq 2 q + q 3 ) + β(q -u) = 0
which is similar to (1) except for the dissipative term which is here nonlinear, because of the geometrical nonlinearity in the membrane. Since the membrane performs large amplitude oscillations, the linear stiffness of the membrane has been neglected in [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators, Part II : Resonance capture[END_REF]. This model reduction yields to explicit formulae for the coefficients β, γ and α 3 with respect to the physical characteristics of the set-up. The natural pulsation of the linear oscillator being ω = πc L , we have 3 Experimental results

β = 2 π S t L V γ = 8 3 h L S t S m ρ m ρ air α 3 = 32 3(1 -ν 2 )π 2 E ρ air c 2 h 3 L R 4 S t S m (3)

Harmonic forced excitation

On figure 4 A, B et C, we can observe a threshold for the appearance of energy pumping. Two stable regimes exist, one without pumping (figure 4 A) when the level of the source is below the threshold, the other with pumping (figure 4 C) when the level is above the threshold. An unstable regime (figure 4 B) also exists around the threshold, the pumping is here intermittently activated. The regime without pumping corresponds to the second nonlinear vibration mode localized on the tube and the regime with pumping corresponds to the first nonlinear vibration mode localized on the membrane as explained in [START_REF] Cochelin | Experimental evidence of energy pumping in acoustics[END_REF]. For the first mode to predominate, a minimum of energy to be brought to the system is required. This minimum corresponds to the energy pumping appearance threshold. This concepts of threshold and mode bifurcation are also well described in [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators, Part II : Resonance capture[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF].

We can also observe the resonance capture phenomenon associated to the bifurcation on figure 5 which is simply a zoom of the figures 4 A(channels 2 and 3) and 4 C(channels 2 and 3). It is clearly visible that during the bifurcation from the mode without pumping to the mode with pumping, a phase shift takes place on the membrane vibrations and that a 1:1 resonance capture appears between the linear system and the nonlinear absorber. The change of localization is also visible : without pumping, vibrations are localized on the tube and with pumping, it is localized on the membrane, meaning that the membrane vibrates instead the air in the tube, a particularly interesting property. Figure 4: Three different experimental osbervations : a regime without pumping (A) for an excitation level below the threshold of pumping appearance, a regime with pumping (C) for an excitation level above the threshold and an unstable intermediate regime (B) where the pumping is intermittent for an excitation level around the threshold. The curves 1, 2 and 3 correspond respectively to the excitation signal, the pressure in the middle of the tube and the velocity of the membrane center.

Noise reduction by energy pumping

In order to clearly observe the reduction of the noise level obtained in the tube resulting from the energy pumping phenomenon, we recorded the signals in two configurations : the first one just before the pumping appearance and the second just after, so that in both cases the level of the source is almost the same. We observe on figure 6 a reduction by a factor of two of the sound amplitude associated to a significant increase of the vibrations of the membrane when pumping is active. For the same excitation level, we have on A the mode without pumping and on B the mode with pumping. The curves 1, 2 and 3 still correspond respectively to the excitation signal, the pressure in the middle of the tube and the velocity of the membrane center.

Free oscillations

The pumping phenomenon is also interesting to rapidely damp the free oscillations after an impulse excitation. Experimentaly, we cannot use a single impulse to start the motion. Instead, we use a harmonic forced excitation at the natural frequency of the set-up. This signal is suddenly turned off, and we observe then the free oscillations of the system. In the case of the curve figure 7 A, we are below the energy threshold for the pumping, and we observe a classical exponential decrease of the pressure in the tube. For a higher value of the initial energy, we have activated the pumping phenomenon as it can be seen in figure 7 B. We can see a linear decrease of the pressure amplitude in the tube, whilst the velocity of the membrane remains important. The unusual fact is that a bigger impulse (above the threshold) is there more rapidely cancelled than a smaller one (below the threshold). It should be noticed that the pumping occurs until the almost complete cancellation of the oscillations in the tube. This observation is of prime interest : we can clearly see an irreversible transfer of energy between the acoustic medium and the membrane. These behaviors are similar to those obtained experimentaly and numericaly on the two d.o.f. mechanical system [START_REF] Gourdon | Energy pumping with various nonlinear structures : numerical evidences[END_REF][START_REF] Mc Farland | Experimental study of nonlinear energy pumping occuring at a single fast frequency[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF].

In the configuration of the figure 7, the pumping threshold is about 125 dB for the sound level in the tube and the level corresponding to the curve B2 is 137 dB. These values are indeed too important for the human hear yet. A part of our future work is then to design the system for lower threshold. : Observation of the free oscillations of the system after turning off the source. On A curves, the initial excitation level is below the pumping threshold. On B curves, the initial excitation level is above the threshold. The curves 1, 2 and 3 still correspond respectively to the excitation signal, the pressure in the middle of the tube and the velocity of the membrane center.

Conclusion

This experimental device permits to show an irreversible transfer from an acoustic medium to a viscoelastic membrane, acting as an energy sink. Many well known results about energy pumping phenomenon have been observed here on an acoustic medium. It is a first step toward a more complete understanding of low frequency noise reduction by using essentially nonlinear oscillators. Parametric optimisation is now scheduled, together with an insight of potential acoustic applications.

Figure 1 :

 1 Figure 1: Two degrees of freedom mechanical system.

Figure 2 :

 2 Figure 2: Principle of the acoustic experimental set-up equivalent to the two d.o.f. mechanical system of figure 1.

Figure 3 :

 3 Figure 3: The experimental set-up. A generator is used to control the loudspeaker. Two measurements are here possible : a microphone measures the pressure in the middle of the tube (easily related to the displacement u of the air at the end of the tube) and a laser vibrometer measures the velocity of the membrane center.

Figure 5 :

 5 Figure 5: Resonance capture : zoom of figure 4 to observe the phases of the signals without pumping (A) and with pumping (B). The continuous line shows the pressure in the middle of the tube and the dash line shows the velocity of the membrane center.

Figure 6 :

 6 Figure 6: Noise reduction in the tube by energy pumping. For the same excitation level, we have on A the mode without pumping and on B the mode with pumping. The curves 1, 2 and 3 still correspond respectively to the excitation signal, the pressure in the middle of the tube and the velocity of the membrane center.

Figure 7

 7 Figure7: Observation of the free oscillations of the system after turning off the source. On A curves, the initial excitation level is below the pumping threshold. On B curves, the initial excitation level is above the threshold. The curves 1, 2 and 3 still correspond respectively to the excitation signal, the pressure in the middle of the tube and the velocity of the membrane center.


P J T Filippi

D Habault

P.-O Mattei

C Maury

This paper is concerned with problems of #uid/structure interaction. The aim is to show how the resonance modes can be used to describe the response of a #uid-loaded structure (displacement of the structure and sound pressure in the #uid) for any kind of excitation (periodic, randomly dependent on time and space, or transient). First, the expression of the response of the system in terms of resonance modes is recalled. To compute the resonance modes, it is necessary to solve a system of equations in the harmonic regime. Several numerical methods are considered and a comparison of their respective e$ciency is presented. Finally, two examples of applications are studied in detail. Namely, a thin plate excited by a turbulent wall pressure and a thin shell excited by a transient acoustic signal. For this last example, some experimental data are compared with numerical results.

INTRODUCTION

The number of publications on #uid/structure interactions over the past three decades is quite impressive. Let us only mention some of the most important books recently published on this topic. The earliest one is due to Junger and Feit [START_REF] Junger | Sound, Structures, and Their interaction[END_REF], the "rst edition of which appeared in 1972. Then two books were published in the 1980s [START_REF] Fahy | Sound and Structural Vibration[END_REF][START_REF] Lesueur | Rayonnement acoustique des structures[END_REF]. These three books provide a very complete overview of the knowledge in the domain and the physics of the phenomena is presented in a classical way which is very easy to follow. A more recent text book [START_REF] Ohayon | Structural Acoustics and Vibration[END_REF] proposes a very rigorous mathematical statement of the problem together with the most recent numerical methods. Let us also mention the CISM Course [START_REF] Habault | Fluid-structure Interactions in Acoustics[END_REF] in which the basic concepts of the #uid/structure interaction phenomena are described. Finally, a very interesting analytical study is due to Crighton [6] who points out the e$ciency of asymptotic techniques.

This paper shows how the resonance modes can be used to describe the response of a #uid-loaded structure (displacement of the structure and sound pressure in the #uid). Most of the results concerning the interaction between a vibrating structure and a #uid, mainly those involving a heavy #uid, show that the response of the structure is governed by 1

The role of the resonance modes in the response of a fluid-loaded structure the #uid-loaded resonance modes. Their role seems essential for large bandwidth excitations (as, for example, turbulent wall pressure) or transient excitations. It is then interesting to develop numerical methods based on the expansion of the response of the system into a series of the #uid-loaded resonance modes. This paper summarizes some results previously obtained by the authors [7}12]. It also includes three original parts.

The "rst one is a presentation of the theoretical development which is more general than the ones given previously (see reference [START_REF] Habault | Fluid-structure Interactions in Acoustics[END_REF], for example). The second one is a comparison of the e$ciency of several numerical methods used to solve the equations of the boundary value problem. The comparison is made on a one-dimensional example. The third one deals with the di!raction of a transient acoustic wave by an axially symmetrical shell (Line-2 shape): numerical predictions are compared with experimental results.

Section 2 is devoted to the description of the equations. The excitation can be harmonically time dependent, transient or random in space and time. First, the response of the system to a harmonic excitation is expanded into a series of eigenmodes which depend on frequency. By using a Fourier inverse transform, it is then possible to express the response of the system to a transient excitation in terms of the #uid-loaded structure resonance modes. These modes do not depend on frequency. When the resonance modes are known, the response of the system to any kind of excitation is obtained. The coe$cients of the series are given in an explicit form, which, in general, cannot be computed numerically. Nevertheless, these coe$cients can also be obtained as solutions of an in"nite system of linear equations which can be solved approximately by a truncation procedure. Section 3 presents several numerical methods for solving the equations corresponding to the harmonic regime and a comparison of their e$ciency on a simple example. The example chosen is a one-dimensional ba%ed plate, in contact with a heavy #uid (water) on one side, and with a vacuum on the other side; the plate is clamped at its boundaries. Two groups of methods are described.

In the "rst group, the equations are transformed into boundary integral equations by using the Green representations for the displacement and the sound pressure. The system obtained is solved by three numerical kinds of approximations. The unknown functions are successively approximated by piecewise constant functions, then by truncated series of orthogonal polynomials (Legendre or Tchebyche! ). A third method consists of approximating the displacement by polynomial functions which satisfy the boundary conditions for the plate displacement, and the acoustic pressure by polynomials. In the second group, the Green representation of the acoustic pressure is still used but the plate equation is written in a variational form. The pressure is approximated by a truncated series of Legendre polynomials, while the plate displacement is approximated by Legendre polynomial functions which satisfy the boundary equations. The pressure/displacement relationship is replaced by a system of &&adapted'' collocation equations; the plate equation is replaced by the corresponding Ritz}Galerkin equations.

Section 4 presents the "rst example: a rectangular ba%ed plate excited by a wall pressure induced by a turbulent #ow. The model of turbulent wall pressure is the one proposed by Corcos [START_REF] Corcos | [END_REF]. The power density spectrum of the plate displacement is computed by solving a sequence of harmonic equations. For a given frequency, the problem is transformed into a system of Boundary Integral Equations and the unknown functions are approximated by truncated series of Tchebyche! polynomials. The coe$cients are solutions of the &&adapted'' collocation equations. The power density spectrum of the plate displacement shows sharp peaks for each resonance frequency of the #uid-loaded structure.

Section 5 presents the second example: a Line 2 shell excited by a transient incident sound wave. The response of the system is expressed as a truncated series of the #uid-loaded structure resonance modes. These modes are calculated by using the variational formulation and polynomial functions satisfying the boundary conditions. The sound pressure di!racted by the shell is calculated and compared to experimental results: the agreement is quite good.

GENERAL STATEMENT OF THE PROBLEM

Consider a thin shell occupying a surface and denote by the exterior domain; is assumed to have everywhere a unit external normal vector n. The shell is made of an elastic material characterized by a density , a Young modulus E and the Poisson ratio ; its thickness is denoted by h. It is immersed in a #uid which extends to in"nity and which is characterized by a density and a sound speed c ; its interior is a vacuum (see Figure 1). For simplicity, it is assumed that all the mechanical and geometrical characteristics of the system are constant. In this section, the superscript && & '' is used for functions which depend on time. The capital letters F, U and P denote vector functions.

The system is excited by a force density F applied to the shell and an incident acoustic pressure pJ G , both of which are zero outside a time interval [0, ¹], F is square integrable in time and space, while pJ G is square integrable in time and on any space domain of "nite dimension (this corresponds to a "nite energy excitation).

Let U be the displacement vector of the shell with components (u , u , u ): the "rst two ones are tangent to , the third one is normal and positive when pointing out to the exterior of the shell. The sound "eld radiated by the shell is denoted by pJ 1

. The equations governing the response of the system have the following form:

AU # hU G #P 1 "F !P G on , p 1 !(1/c )pJ ( 1 "0 in , uJ ( #Tr * L pJ 1 "!Tr * L pJ G on . ( 1
)
Here the matrix A is the partial derivative operator which describes the shell properties. The notation Tr f stands for the limit value of the function f when the current point in tends to a point on : this notation is necessary when f has a discontinuity across or is expressed by an integral which becomes hyper-singular on (this occurs with layer potentials). P 1 is the vector (0, 0, Tr pJ 1

) and P G is the vector (0, 0, Tr pJ G ). The notation f $ stands for the second partial derivative of f with respect to time. These equations are completed by initial conditions (for example, every function and its "rst time derivative is zero for t(0), by boundary conditions and, if needed, by continuity conditions on U . An outgoing wave condition is also added for pJ 1 . In the domain , the acoustic pressure "eld pJ 1 can be represented by a boundary integral in terms of a layer density N J , that is,

pJ 1 "K I N J , (2)
where K I is an integral operator. In general, its kernel will be any linear combination of the free"eld Green kernel of the wave equation and its normal derivative (see its Fourier transform in equation ( 8)). For simple geometries (ba%ed plate, for example), use can be made of the Green function which satis"es the Neumann boundary condition on .

The values of pJ 1 and of its normal derivative on the shell surface can be written as

Tr pJ 1 "K I N J , Tr * L pJ 1 "K I N J , (3)
where K I and K I are related to K I in a very simple way. The system of equations ( 1) is then replaced by

AU # hU G #K M I N J "F !P G on , uJ ( #K I N J "!Tr * L pJ G on . (4)
Let us note that the "rst equation is a vector equation where K M I N J is the vector (0, 0, K I N J ). The second equation is scalar. All the unknown functions are de"ned on the shell surface only. These equations can also be written in a variational form (which corresponds to the energy balance)

A(U , V )# dt h[U G #K M I N J ] ) V * d " dt [F !P G ] ) V * d , dt [ uJ ( #K I N J ] I * d "! dt Tr * L pJ G I * d , (5)
where V and I are test functions, A(U , V ) is the bilinear form corresponding to the expression of the potential energy of the shell, ( )* denotes a complex conjugate and U ) V is the scalar product of the two vectors U and V. In the following, all integration elements will be omitted unless there is any ambiguity.

HARMONIC REGIME AND EIGENMODES SERIES

The basic idea developed in this subsection and in the next one is the same as that used by Morse and Ingard [START_REF] Morse | Theoretical Acoustics[END_REF] (section 9.5 on Room Acoustics) for the evaluation of the acoustic response of a room to either a harmonic or a transient source. They consider a room with absorbing walls and make several assumptions. Though the physical system is not conservative, for any excitation frequency the existence of eigenmodes and eigenvalues which are frequency dependent is assumed; the response of the room to any source is expanded into a series of these eigenmodes. The response of the room to a transient excitation is then determined by an inverse Fourier transform which is calculated by the residues method under the following hypotheses: the eigenmodes and eigenfrequencies depend analytically on the frequency; the coe$cients of the harmonic response expansion have poles only (no branch cuts) which correspond to the resonance frequencies (free oscillations) of the room. We do not know any general mathematical justi"cation of this theoretical approach. But as long as an analytical solution exists (separation of variables), this approach can be applied. From a physicist's point of view, what is true for a parallelepipedic enclosure should remain true for any other shape of practical interest.

For the present problem, the vibro-acoustic response of a #uid-loaded shell, it can be proved that the forthcoming developments are valid in very simple situations. We, thus, assume that they remain valid for any situation. We will precise the assumptions made at each step.

Let f be the time Fourier transform of a function f I de"ned as it is usual in acoustics by

f ( )" > \ f I (t)e SR dt. (6)
Equations ( 5) become

A(U, V)# [! hU#K M S N ] ) V*" [F!P G ] ) V*, [! u #KS N ] *"! Tr * L p G *, (7)
where K M S and KS are the Fourier transforms of the operators K M I and K I . They depend on the angular frequency . More precisely, the integral representation of the acoustic pressure "eld, its value and the value of its normal derivative on can be written as (8) where G(M, M)"!exp(i r(M, M)/c )/(4 r(M, M)) is the free"eld Green kernel of the Helmholtz equation which satis"es the Sommerfeld condition corresponding to the time dependence, is a function of with non-zero imaginary part and such that (! *)" *( ), and the symbol Pf denotes the "nite part of the hyper-singular integral. With this choice, the operator KS has an inverse +KS ,\ for any real frequency, that is the function N can be calculated for any given and Tr p 1 . Finally, replacing the unknown function N by :

p 1 (M)"KS N " [G(M, M)# * L +Y G(M, M)] N (M) dM, Tr p 1 (M)"KS N "! N (M) 2 # [G(M, M)# * L +Y G(M, M)] N (M) dM, Tr * L p 1 (M)"KS N "! N (M) 2 #Pf * L + [G(M, M)# * L +Y G(M, M)] N (M) dM,
" N / , (9)
one gets

A(U, V)! h U! h K M S ) V*" [F!P G ] ) V*, [!u #KS ] *"! 1 Tr * L p G *. ( 10
)
Let us de"ne the eigenmodes and eigenvalues of equation (10). Their existence, which can be proved in very simple situations, is assumed for the general case. Introducing the notation U < L "(u( L , u( L , u( L ), the eigenmodes (U < L , L L ) and eigenvalues L satisfy the following system of homogeneous equations:

A(U < L , V)! L U < L ! h K M S ( L ) V*"0, [!u( L #KS ( L ] *"0. ( 11
)
From the last equation, ( L can be expressed as a function of u( L :

( L "YS u( L , ( 12
)
where YS is the symbolic inverse of KS . The system of equations (11) reduce to one equation only

A(U < L , V)! L U < L ! h K M S (YS u( L ) ) V*"0. (13)
The eigenmodes and eigenvalues depend on the angular frequency. It can be shown [START_REF] Maury | The`se de meH canique*speH cialiteH : Acoustique et dynamique des vibrations[END_REF] that the eigenmodes satisfy the following orthogonality relationship:

A(U < L U < * K )" L U < L ! h K M S L L ) U < K "0 if mOn "N L ( ) i f m"n. ( 14
)
Here, N L can be seen as playing a role equivalent to the norm of U < L . The displacement U, solution of equations ( 10) is sought as a series of the eigenmodes:

U" L L U < L . ( 15
)
By introducing this series and using the formal inverse of KS , the system of equations ( 10) is replaced by the following equation:

L L A(U < L , V)! h U < L ! h K M S (YS u( L ) ) V* " [F!P G #K M S (YS Tr * L p G ] ) V*. (16)
To determine the coe$cients of the expansion of U, this equation is written for V"U* K . An explicit form of the operator YS is not needed. Indeed, the Green formula applied to the functions

G "KS(YS Tr * L p G ) and K K "KS ( L (17)
leads to the following equality:

KS(YS Tr * L p G )u( K " Tr * L p G Tr K K . ( 18
)
This result together with the orthogonality relationship provides the following expression for the coe$cients L :

L "! L N L ( )( h! L ) [(F!P G ) ) U < L #Tr * L p G Tr K L ]. (19)
Finally, the acoustic pressure radiated by the shell is given by the series

p 1 " L L p( L #p , ( 20
) with pL L " KS ( L and p , "!KS(YS Tr * L p G ).
The sum p G #p , satis"es a homogeneous Neumann condition on the shell surface, that is p , is the pressure "eld re#ected by a perfectly rigid structure. The result is summarized as follows: (21) where all the quantities involved, considered as functions of the complex variable satisfy the following relationship:

U" L L U K L , p 1 " L L K L #p , , L "! L L N L ( )( h! L ) , L " [(F!P G ) ) U < L #Tr * L p G Tr K L ],
f (! *)"f *( ) (22)
for any real excitation (transient force and incident acoustic pressure).

TRANSIENT REGIME AND RESONANCE MODES

The transient response of the system is obtained by taking the inverse Fourier transform of the harmonic displacement and pressure "eld. This leads to a representation of the solution in terms of the resonance modes of the #uid-loaded structure (free oscillation modes).

The existence of the resonance modes and of the resonance frequencies has been proved for three-dimensional elasticity [START_REF] Bardos | [END_REF]. It is assumed that the results of this reference are still valid for the thin-body approximate equations of elasticity.

The resonance modes (U L , p L ) of the structure/#uid system and the resonance angular frequencies L are de"ned by

A(U L , V)! L h U L ! h K M SL L ) V*"0, [!u L #KSL L ] *"0, p L " L KSL L . ( 23
)
The resonance frequencies L are related to the eigenvalues L de"ned in the previous paragraph by

L h" L ( L ). ( 24
)
It has been shown that, for any nO0, this equation has two solutions that we denote L and \L . They have the following property [START_REF] Bardos | [END_REF]:

L " L !i L with L '0, L '0, \L "! * L . ( 25
)
The inverse Fourier transforms to be calculated have the following form:

F I (t)" 1 2 > \ f ( ) N L ( )( h! L ) e\ SR d . ( 26
)
The use of the method of residues immediately leads to

F I (t)"!iY(t) f ( L ) N L ( L )[2 L h! L ( L )] e\ SLR! f *( L ) N* L ( L )[2 * L h! * L ( L )] e S * L R , ( 27
)
where Y(t) is the Heaviside step function and L ( ) is the derivative of L with respect to ; use has been made of the result L (! *)"! * L ( ) which is deduced from the property of L ( ) given in equation [START_REF] Durant | et de la re & ponse vibro-acoustique d'une conduite sollicite& e par un e& coulement interne[END_REF]. The function F I (t) is real valued. Thus, the transient response of the system has the expression:

U (t)"iY(t) L L ( L ) L ( L ) N L ( L )[2 L h! L ( L )] U L e\ L R ! * L ( L ) * L ( L ) N* L ( L )[2 * L h! * L ( L )] U* L e L R e\O L R, pJ (t)"pJ G (t)#pJ , (t)#iY(t) L L ( L ) L ( L ) N L ( L )[2 L h! L ( L )] p L e\ L R ! * L ( L ) * L ( L ) N* L ( L )[2 * L h! * L ( L )] p* L e L R e\OLR, ( 28
)
where Y is the Heaviside function. This kind of expression is used in example 2 (section 5). This result is based on the following hypotheses. The eigenmodes and the eigenvalues depend analytically on the frequency (this is reasonable because they are solutions of an equation which depends analytically on the frequency). As a consequence, the factors N L ( ) have the same property; furthermore, it is assumed that they have no zero.

Let us make a "nal comment on the expansions in resonance modes. By applying the time Fourier transform to expression (28), one gets a representation of the response of the system to a harmonic excitation in terms of resonance modes:

U( )"i L L ( L ) L ( L ) N L ( L )[2 L h! L ( L )] 1 i( L ! )# L U L # * L ( L ) * L ( L ) N* L ( L )[2 * L h! * L ( L )] 1 i( L # )! L U* L , p( )"p G ( )#p , ( )#i L L ( L ) L ( L ) N L ( L )[2 L h! L ( L )] 1 i( L ! )# L p L # * L ( L ) * L ( L ) N* L ( L )[2 * L h! * L ( L )] 1 i( L # )! L U* L . ( 29
)
This last expression shows clearly that the response of the system is maximum when the excitation frequency is equal to the real part of any resonance frequency. Such a property is not pointed out by the eigenmodes series representation of the solution.

REMARKS

(1) From a purely theoretical point of view, the series representations (28) and (29) are very attractive because explicit expressions of the coe$cients are given. But, in practice, they cannot be calculated easily mainly because they involve the derivative of the eigenvalues with respect to the angular frequency.

When the #uid is a gas, its in#uence on the vibrations of the structure is small and a perturbation method can be used to derive an approximation of the resonance modes and frequencies of the #uid-loaded structure from its in vacuo modes and frequencies. An approximation of expressions (28) and (29) can then be obtained. In general, the coe$cients of the expansions are computed by solving a truncated form of an in"nite system of algebraic linear equations.

(2) The main di$culty is to calculate the #uid-loaded resonance modes. There exists a few cases for which a method of separation of variables is available. In general, only numerical approximations can be obtained. It is then necessary to have an e$cient computer program for solving the system of equations ( 23) which correspond to the steady state equations for any complex angular frequency.

For this reason, in the next section several numerical methods to solve equations (23) are examined for the simple example of a one-dimensional ba%ed plate excited by a point harmonic source. The convergence of these methods has already been proved in theoretical papers (see references [START_REF] Kress | Linear Integral Equations[END_REF]18], for example). Here our point is to develop numerical comparisons. The results obtained are discussed in order to provide criteria for determining the values of the parameters corresponding to each method.

(3) Finally, it must be mentioned that the representation of the solution in terms of resonance modes is much more interesting for transient regimes. For harmonic or random excitations, a direct numerical method ("nite elements, boundary elements, etc.) is often much less time consuming. But for transient excitations, the series of resonance modes seems to be the best way to describe the successive wavefronts which are experimentally observed. For example, in reference [START_REF] Bardos | [END_REF], the contributions of the resonance modes are easily interpreted as surface waves.

COMPARISONS BETWEEN SOME NUMERICAL METHODS

The example chosen in this section is a one-dimensional ba%ed plate excited by a harmonic source. Two main methods are described. In the &&fully integral'' method, the acoustic pressure and the plate displacement are represented by integrals based on the use of the Helmholtz and plate Green kernels. In the &&mixed integro-di!erential'' method, the acoustic pressure is represented by a boundary integral, while the plate displacement is considered as the solution of a di!erential equation.

STATEMENT OF THE EXAMPLE

Consider a one-dimensional plate which occupies the segment "(!¸/2(x(# ¸/2) of the y"0 axis and which is clamped at both ends. The remaining part "("x"'#¸/2) is a perfectly rigid ba%e. The half-space D"(y'0) is occupied by a #uid, while the other half-space is a vacuum. The characteristics of the plate are: thickness h, density , Young's modulus E, the Poisson ratio . The characteristics of the #uid are: density , sound speed c . The plate is excited by a harmonic unit point force located at x"S; the excitation angular frequency is ; and there is no acoustic source.

Let w and p be the plate displacement and the acoustic pressure respectively. They satisfy the following system of equations:

D(*w/*x)! h w#P" 1 on , p#kp"0 in D, Tr (*p/*y)" w on 0 o n w(!¸/2)"w(¸/2)"w(!¸/2)"w(¸/2)"0, with D"Eh/12(1! ) , k" /c , P"Tr p. ( 30
)
The uniqueness of the solution is ensured by adding a Sommerfeld condition on p. Let G be the Green function of the Helmholtz equation which satis"es a Neumann condition on 6 and the Sommerfeld condition. The acoustic pressure is expressed in terms of the plate displacement by p" Gw.

(31) Using this expression, the system of equations (30), is replaced by a system of integro-di!erential equations on :

D(*w/*x)! h w#P" 1 on ,
Gw!P"0 on ,

w(!¸/2)"w(¸/2)"w(!¸/2)"w(¸/2)"0. ( 32
)
The variational form of these equations is given by

D *w *x *v* *x ! h wv*#Pv* "v*(S) on , Gw!P *"0 on , (33)
where v and are test functions (see, for example, reference [START_REF] Ohayon | Structural Acoustics and Vibration[END_REF]). Let be the Green kernel of the plate equation which satis"es the Sommerfeld condition corresponding to outgoing waves. A Green's formula applied to and w gives a representation of w in terms of the excitation force, the acoustic pressure and boundary sources. The system of equations ( 32) is thus replaced by the following system of integral equations:

w# * P# * [u> * !u\ \* #u> * !u\ \* ]" * 1 on ,
Gw!P"0 on , (34)

w(!¸/2)"w(¸/2)"w(!¸/2)"w(¸/2)"0,
where * f stands for the convolution product of the plate Green kernel by the function f and ? "* ? /*x is the derivative with respect to x of the Dirac measure located in x"a; the scalars u $ are the values of the third derivative of w at x"$¸/2; the scalars u $ are the values of the second derivative of w at x"$¸/2. The boundary conditions provide the four additional integral equations required to determine the two unknown functions and the four boundary sources.

In the next subsections, two families of numerical methods are studied. The "rst family solves equations (34): two families of approximation functions are used and the system is solved by a collocation method. We call it the boundary integral/collocation method. The second family solves equations (33). The unknown functions are approximated by polynomial functions. The di!erential equation is approximated either by Ritz}Galerkin equations or by collocation equations, while the integral equation is approximated by collocation equations. We call it the mixed diwerential-boundary integral/Ritz-collocation method. These methods are quite classical. The aim of this section is to compare their e$ciency and provide some practical rules of convergence for the particular case of a heavy #uid (water).

THE BOUNDARY INTEGRAL/COLLOCATION METHOD

The simplest approximation of the solution of equation ( 34) is provided by piecewise constant functions. A more re"ned approximation is obtained by using orthogonal polynomials.

Approximation by piecewise constant functions

The interval is divided into sub-intervals G centered at the points x G . It must be remarked that the plate displacement and the acoustic pressure have not the same wavelength. Thus, two sets of sub-intervals are de"ned: B G (i"1, 2 , Nd) for the displacement and N G (i"1, 2 , Np) for the pressure. The "rst equation ( 34) is written at the centers of each interval B G , while the second one is written at the centers of each interval N G . This "nally provides (Nd#Np#4) equations when the four boundary conditions are added.

The functions w and P are approximated by piecewise constant functions with values w G and P G on G . The accuracy of the approximations depends on two main parameters. The "rst one is, of course, Nd and Np which must be large enough so that the space variations of the unknown functions are correctly described. The second parameter is the accuracy of the numerical integration of the kernels and G over each sub-interval G .

Approximation by orthogonal polynomials and adapted collocation

The unknown functions are approximated by truncated series of orthogonal polynomials ; G (we have tested Legendre and Tchebyche! polynomials):

w+ ,B G w G ; G , P+ ,N G G ; G .
The "rst equation of equation ( 34) is written at the zeros of the polynomial ; ,B> and the second one at the zeros of the polynomial ;

,N> . This leads to a system of equations that are called here &&adapted collocation equations''. The interest of choosing these collocation points is that the equations obtained are equivalent to Ritz}Galerkin equations which are known for giving a much more accurate result (see, for example, references [7] or [START_REF] Maury | The`se de meH canique*speH cialiteH : Acoustique et dynamique des vibrations[END_REF]).

It is also possible to choose for the ; G a combination of orthogonal polynomials which satisfy the boundary conditions. Thus, the four equations which express the boundary conditions and require a rather signi"cant computing time, are no more necessary. This results in a signi"cant computing time economy.

THE MIXED DIFFERENTIAL-BOUNDARY INTEGRAL/RITZ COLLOCATION METHOD

In this section, the acoustic pressure in equations ( 33) is approximated by a sum of normalized Legendre polynomials G

(x)"A G P G (2x/¸) (A G is the norm of P G ): P+ ,N G N G G . ( 35
)
The displacement is approximated by a sum of polynomial functions. These polynomial functions are linear combinations of Legendre polynomials chosen such to satisfy the boundary conditions. The displacement approximation and the approximation functions are de"ned by

w+ ,B G w G = G , = G (x)"B G P G 2x ¸ ! 4i#10 2i#7 P G> 2x ¸ # 2i#3 2i#7 P G> 2x ¸ , (36)
where B G is a normalizing factor. The function = G is a polynomial of degree (i#4). It seems reasonable to adopt the same degree of polynomial approximations for both the pressure and the displacement: thus Np"(Nd#4).

These approximations are introduced into equations (33). The test functions are chosen equal to the approximation functions, that is the equations are written for

v"= G , i"1, 2, 2 , Nd, " G , i"1, 2, 2 , Np. ( 37
)
The "rst equation ( 33) is replaced by the corresponding Ritz}Galerkin equations, while the second equation is replaced by the set of adapted collocation equations:

,B G w G D *= G *x *= H *x ! h = G = H # ,N I N I I = H "= H (S), j"1, 2, 2 , Nd, ,B G w G G= G (X P )! ,N I N I I (X P )"0, r"1, 2, 2 , Np, (38)
where the X P are the zeros of the function ,N> . The main advantage of this formulation is that the parameters D and h can be functions of the variable x. In the "rst equation (38) the numerical evaluation of the integrals requires a rather large amount of time.

For constant plate parameters, the integrals involved in the left side of the "rst equations can be computed exactly by using a Gauss}Legendre procedure. In this case and if the excitation force is a square integrable function F(x), an adapted collocation can be used and the system is replaced by

,B G w G D *= G *x (> O )! h = G (> O )# ,N I N I I (> O )"F(> O ), q"1, 2, 2 , Nd, ,B G w G G= G (X P )! ,N I N I I (X P )"0, r"1, 2, 2 , Np, (39)
where the > O are the zeros of the polynomial ,B> . This approximation can be improved by using more collocation points in both of the two equations and solving the rectangular system thus obtained by a least-squares method.

In the last section, the transient vibro-acoustics response of a Line-2 is studied. Though this shell has a constant thickness and is made of a homogeneous material, the elasticity equations have variable coe$cients. Thus, the approximation equations which have been adopted are similar to those described in this subsection.

NUMERICAL COMPARISON BETWEEN THE TWO FAMILIES OF APPROXIMATIONS

The example studied in this section is a one-dimensional ba%ed plate, immersed in water and excited by a harmonic point force. The response of the system is computed by the methods presented in the previous section. For numerical comparisons, three plates are examined. Their characteristics are summarized in Table 1. The &&coupling factor'' (CF) is de"ned as the ratio / h. The length of each plate is ¸"1 m. The #uid (water) is characterized by "1000 kg/m and c "1500 m/s. These three examples present a particular interest from a numerical point of view. The ratio h/D in the "rst two plates have very close values but the coupling factor is di!erent. This means that plates 1 and 2 have very close values for their in vacuo eigenfrequencies but they have di!erent resonance frequencies in water (see below). Plates 2 and 3 correspond to the same material but di!erent thicknesses. Then, comparisons between plates 1 and 2 point out the in#uence of the coupling factor and comparisons between plates 2 and 3 point out the in#uence of the thickness of the plate. The "rst eigenfrequencies of the plates (in vacuo) and the "rst resonance frequencies in water are:

Eigenfrequencies:

Plate The two families of methods described in the previous section have been applied. Method I is the boundary integral/collocation method. The approximation functions are piecewise constant functions. Nd and Np are the numbers of constants used to approximate the displacement and the sound pressure respectively. The two series of collocation points are equally spaced on the plate. As seen before, this leads to a linear system of order (Nd#Np#4). In method I, the collocation functions are orthogonal (Legendre or Tchebyche! ) polynomials and the two series of collocation points are the zeros of the polynomials Nd#1 and Np#1 respectively.

Method II is the mixed di!erential-boundary integral/Ritz-collocation method. The displacement is approximated by polynomial functions and the sound pressure by Legendre polynomials. Nd and Np are the numbers of functions used to approximate the displacement and the sound pressure respectively. We have chosen to approximate the displacement and the pressure by polynomial functions of the same maximum degree. This corresponds to Np"(Nd#4). As mentioned above (section 3.3), if the plate excitation is a square integrable function, the Ritz equations can be replaced by a set of adapted collocation equations (method II). The frequencies studied range from 56 to 1500 Hz. For each frequency, the convergence has been studied by varying the values of (Nd, Np). Three types of curves have been computed: the displacement on the plate, the sound pressure on a line parallel and very close to the plate (distance"¸/40), and the sound pressure radiated at a distance R"1 m from the plate.

The convergence of each method is evaluated from comparisons on the displacement curves (on a linear-scale) only. This corresponds to the most strict conditions. In most cases, the convergence on the other two curves (sound pressure at ¸/40 and at R"1 m) is reached for lower values of (Nd, Np). Let us mention that the values given in Tables 2 and3 are not exactly the lowest values which give an accurate displacement. For each plate and frequency, 10}12 couples (Nd, Np) have been tested. The values given in the tables correspond to the lowest values tested which gave the right curve (computed with large values of (Nd, Np)) with a discrepancy less than 1%.

The results obtained with method I are summarized in Table 2. For each plate and each frequency, Table 2 gives the values (Nd, Np) and the number M of zeros of the displacement curve. In the convergence tests, the values of Np have been chosen smaller or at most equal to Nd since the most classical criterion for collocation methods applied to integral equations is simply related to the wavelength in the propagation medium. It says that the number of collocation points must be such that the length of the sub-elements is equal or less than one-sixth of the wavelength. In the examples chosen, the wavelength in the #uid is much smaller than the wavelength in the plate. For example, at 500 Hz, the wavelength in water is equal to 3 m, the wavelength in plate 1 and 2 is 0)45 m, the wavelength in plate 3 is 0)25 m.

Table 2 shows that this criterion does not apply so simply. First, the best results have been obtained for equal values of Nd and Np. Furthermore, the numbers Nd and Np clearly depend not only on the wavelength but also on the coupling factor.

Although the wavelength is the same in plates 1 and 2, Nd is larger for plate 2 for which the coupling factor is approximately three times larger than the coupling factor for plate 1.

To emphasize the role of the coupling factor, let us mention the results obtained in air. For the same calculation for plate 3, only 10}01 is needed at 56 Hz and 12}07 at 120 Hz, the coupling factor is equal to 0)16 m\ in this case.

The same curves have been obtained with method I, with similar values of (Nd, Np). It does not seem that using orthogonal polynomials provides any improvement, from a numerical point of view. On the contrary, for equal values of (Nd, Np), this requires a larger computing time since it includes the computation of integrals of products of polynomials and Hankel functions.

Figures 2 and3 show two examples of convergence for the displacement on plate 2, for two frequencies 56 and 1500 Hz.

The results obtained with method II are summarized in Table 3. For each frequency and each plate, Table 3 gives the values Nd and Np"(Nd#4). M has of course the same values as in Table 2.

It can be seen that the values of Nd mainly depend on the number of maxima of the displacement. In most cases, 2M)Nd)4 M. For both plates 1 and 2, values of the ratio h/D are very close to each other; thus, at each frequency, the value of Nd can be chosen of the same order of magnitude for both plates. It is larger for plate 3 for which the wavenumbers and then M are larger. For plate 3, we do not give any result at 1500 Hz: it appeared that the system of equations becomes unstable unless the working precision is increased (for example with a Mathematica program, the regular precision of 16 digits is not su$cient): nevertheless, by taking precautions, it has been possible to suppress these instabilities and obtain reliable results.

Figures 4 and5 show the convergence of the method for plate 2 at 56 and 1500 Hz. Figures 6 and7 present a comparison between the curves obtained for plate 2 with methods I and II at 56 and 1500 Hz respectively. The two methods do not give exactly the same result. The di!erences on the maxima is about 2% at 56 Hz, and 5% at 1500 Hz: this is absolutely not signi"cant, the di!erence induced on the pressure "eld being much less than 0)1 dB. Nevertheless, it is necessary to give an explanation of this discrepancy. It is of course possible to prove that both approximation methods converge to the exact solution. But, in practice, due to rounding errors, the numerical results do not converge to the exact solution; furthermore, the rounding errors depending on the numerical method, a perfect agreement between various numerical approximations cannot be expected.

On a few examples, method II has also been tested. It appears that, for the same choice of (Nd, Np), the results have the same accuracy, but the computation time is reduced.

Finally, Figures 8 and9 present the directivity patterns of the plates at 56 and 1500 Hz, calculated with method II. It has been shown that for both methods, the convergence does not depend only on the wavelengths in the #uid and in the plate. The coupling factor is also an important parameter. The comparison of Tables 2 and3 provides an interesting result. The rules of convergence are di!erent with methods I and II. With method I, the e!ect of the coupling factor is clearly essential when the wavenumbers are equal (as for plates 1 and 2). With method II, the values of the wavenumber in the plate (or the number M of maxima) is the main parameter.

RESPONSE OF A THIN PLATE TO A TURBULENT WALL PRESSURE EXCITATION

The aim of this section is the prediction of the response of a simple structure*a two-dimensional ba%ed rectangular plate*induced by a turbulent #ow, that is the determination of the various power density spectra (displacement and pressure "elds). The calculation of these spectra is based on the solution of the #uid-loaded plate equation for a continuous set of harmonic excitations. The method used here is based on the expansion of the unknown functions in series of Tchebyche! polynomials and adapted collocation equations, as described in section 3.2.2.

The plate is made of purely elastic steel characterized by: Young's modulus E"2)0;10 Pa, Poisson ratio "0)3, density "7800 kg/m; its size is 1)0;0)7 m; with thickness h"0)05 m.

One of the two half-spaces is occupied by water, with density 1000 kg/m and sound velocity 1500 m/s. The other half-space is a vacuum.

The plate is excited by a turbulent #ow parallel to its longest sides. There is no acoustic source. The acoustic #uid loading of the plate is modelled as if the #uid was at rest: indeed, a few numerical experiments have shown that, for small Mach numbers, the #uid velocity has a quite negligible in#uence on the acoustic pressure on the plate.

MODEL USED FOR THE TURBULENCE AND RESPONSE OF THE PLATE TO THE CORRESPONDING WALL PRESSURE

The turbulent wall pressure excitation, which is a space}time random process, is characterized by a cross-power density spectrum [START_REF] Crighton | Modern Methods in Analytical Acoustics[END_REF] N (x, y; x, y; ) where is the angular frequency. Among the many models available in the literature (see for example references [20] or [START_REF] Blake | Mechanics of Flow-Induced Sound and Vibration[END_REF]), we have chosen the one proposed by Corcos who adopts the following form for the wall pressure:

N (x, y; x, y; )" N ( ) e\?VSV\VY3A e\? W SW\WY3A e S W\WY 3A (40)
with W "0)1, V "7 W and ; A "0)7;

. Following reference [START_REF] Durant | et de la re & ponse vibro-acoustique d'une conduite sollicite& e par un e& coulement interne[END_REF], the spectrum

N ( ) is de"ned by log C ( )"!5)1!0)9 log f C !0)34 (log f C )!0)04(log f C ) with f C " /2 ; , q "1/2 ; , N ( )"(q /; ) C ( ), (41)
where "0)10 m is the turbulent boundary layer thickness.

As shown in many papers (in particular in reference [START_REF] Filippi | Uncertainty Modeling in Finite Element, Fatigue and Stability of Structures[END_REF]), the cross-spectral density S S (M, M; ) of the plate displacement is related to the cross-spectrum density of the excitation by

S S (M, M; )" u(Q, M; ) N (Q, Q; ) u*(Q, M; ) dQ dQ, (42)
where u(Q, M; ) is the response of the #uid-loaded plate to a harmonic unit point force / and u* is the complex conjugate of u. The numerical problem is the determination of the plate displacement u for a harmonic excitation.

MAIN STEPS OF THE COMPUTATION AND RESULTS

The computation of u(M, Q; ) was made with the numerical method presented in section 3.2.2. The acoustic pressure on the plate surface is described by the Green representation corresponding to a #uid at rest. The Green representation of the plate displacement is used: it involves boundary layer integrals to account for the boundary conditions. The Green representation of the plate displacement, the continuity condition between the plate displacement and the normal component of the #uid particle displacement, and the boundary conditions provide a system of four integral equations.

The unknown functions which depend on one variable*the two layer densities*are expanded into truncated series of Tchebyche! polynomials. The unknown functions which depend on two variables*the plate displacement and the acoustic pressure on the plate surface*are expanded into truncated series of products of two Tchebyche! polynomials. The coe$cients of these expansions are evaluated by an adapted collocation method. A detailed presentation of this speci"c case is given in reference [7].

The last step is the evaluation of the power density spectrum of the displacement. This requires the computation of quadruple integrals. But, due to the variable separation of both the plate displacement representation and the power density spectrum of the excitation, the calculation is reduced to the evaluation of the product of two double integrals. Figure 10 presents an example of the response of the plate excited by a turbulent boundary layer. The power density spectrum of the plate velocity S T " S S has been calculated for two di!erent conditions. The continuous curve is the response of the water-loaded plate. It is compared to the response of the in vacuo plate excited by the same wall pressure. This clearly shows the #uid-loading in#uence. The two curves are obtained for x"0)9¸V and y"0)9¸W. With this location, all the modes of the plate are visible. The #uid loading has two consequences which can be observed on these curves: the added mass e!ect, that shifts the peaks toward the low-frequency domain, and the damping e!ect, that reduces their amplitude. The positions of the amplitude maxima are compared to the resonance frequencies of the #uid-loaded plate (see Table 4). One can see that the maxima of the response of the plate occur for frequencies which are very close to the real part of the resonance frequencies; their amplitudes get smaller as the damping of the resonance modes increases. For example, the response of the plate close to 82 and 90 Hz presents a di!erence of more than 20 dB. This could be expected simply by looking at the resonance frequencies: at 90 Hz the resonance frequency has an imaginary part 100 times greater than the one at 82 Hz. The mode that corresponds to 90 Hz is signi"cantly damped by the #uid loading. One can also remark that the resonance modes which present a high damping are modes with a maximum amplitude at the center of the plate (this occurs for the "rst, fourth, eighth, etc., modes).

This example clearly shows that, for an excitation with a wide frequency band, the response of the system structure/#uid is governed by the #uid-loaded modes. It also suggests that a re"ned mesh is required around each resonance frequency, while a rather large frequency mesh or an interpolation algorithm between two successive resonance frequencies is su$cient. In our opinion, the use of the resonance modes representation of the structure response is certainly one of the best methods, if not the best. The example of the next section seems to con"rm the assertion.

DIFFRACTION OF A TRANSIENT ACOUSTIC WAVE BY A ¸INE-2 SHELL

The shell Line-2 is composed of three elementary thin shells , and . The elements and are two identical hemi-spherical end-caps which close the extremities of a cylindrical element . They all have the same mean radius R and the same thickness h which is assumed to be a few percents of R; the length of the cylindrical part is 2¸.

The three elements are made of the same material, characterized by a density Q , a Young modulus E and the Poisson ratio .

A co-ordinate system is associated with each element (see Figure 11). The shell is immersed in a #uid, water, extending to in"nity and is excited by an incident transient wave.

As described in section 2.2, the response of the system is sought as a series of the #uid-loaded resonance modes of the shell and of the corresponding radiated acoustic pressure. The resonance modes are computed with the method described in section 3.3. The details concerning this example are presented in references [START_REF] Maury | Transient acoustic di!raction and radiation by an axisymmetrical elastic shell: a new statement of the basic equations and a numerical method based on polynomial approximations[END_REF][START_REF] Maury | The`se de meH canique*speH cialiteH : Acoustique et dynamique des vibrations[END_REF].

MAIN STEPS OF THE NUMERICAL PROCEDURE

The "rst step is to compute the resonance modes. This requires one to solve the homogeneous equations governing the harmonic regime.

Because the structure has an axis of symmetry, it is possible to expand all the unknown quantities*the shell displacement (three components), the layer density and the corresponding di!racted acoustic pressure*into Fourier series of the angular variable .

The angular harmonics of the shell displacement components and of the layer density satisfy a system of integro-di!erential equations of one variable only, the curvilinear abscissa along a meridian of the shell. Because of the spherical end-caps, variable coe$cients occur. The shell displacement is approximated by polynomial functions (linear combinations of Legendre polynomials) which satisfy the boundary conditions at the poles and the continuity conditions at the junctions of the spherical parts of the shell and the cylindrical part. The layer density is approximated by a truncated series of Legendre polynomials. Use is made of Ritz}Galerkin equations for the shell equation and of adapted collocation equations for the continuity relationship between the normal component of the shell displacement and of the #uid particle displacement.

A very straightforward method is used to calculate the transient response of the shell. For each angular harmonic, each component of the shell displacement is expressed as a linear combination of the corresponding component of the modal resonance displacements; the layer density is expressed as a linear combination of the modal resonance layer densities. These four expansions, which are considered as independent, are introduced into the variational form of the governing time-dependent equations. Using the modal resonance displacement components and the modal resonance layer densities as test functions, one obtains a system of linear algebraic equations which provides the unknown coe$cients.

COMPARISON BETWEEN NUMERICAL PREDICTIONS AND EXPERIMENTAL RESULTS

A Line-2 shell, made of an aluminum alloy, has been used for experiments. Its radius is 0)020 m, its total length is 0)080 m and its thickness is 0)001 m. Its mechanical characteristics are: density"2611 kg/m, Young's modulus 0)81;10 Pa, the Poisson ratio"0)30. It was immersed in a large water tank with anechoic boundaries. An acoustic transducer, located on the symmetry axis of the target, was used for the generation of a short signal with a spectrum centered around 250 kHz (four oscillations, lasting 0)2 ms). A unique transducer was used for the sound emission and the recording of the re#ected wave. The re#ected wave in the backward direction was recorded 1 m away from the shell.

In Figure 12, the experimental signal is compared with the predicted one, for a total time interval of 2)0 ms, that is 10 times the duration of the incident wave. It must be noticed that the re#ected wave is very sensitive to the relative positions of the transducer and of the target: a small error in the orientation of the target can change the di!racted wave by a rather large amount. Nevertheless, the agreement between the two curves is very good.

Though the re#ected signal was computed as a whole, the specular re#ection and the successive wave packets, which correspond roughly to creeping waves having travelled several times around the target, appear clearly. This shows the strong e$ciency of the expansion of the di!racted acoustic transient pressure in terms of the resonance modes.

CONCLUSION

The aim of this paper was to point out the e$ciency of the resonance modes of the #uidloaded structure to describe the response of a coupled system structure/#uid to any excitation and more speci"cally to a transient excitation.

In all the examples presented, the #uid is water, because, in a heavy #uid, the resonance modes are far from the in vacuo modes and cannot be easily deduced from them, as, for example, by a perturbation method.

The resonance modes are obtained by solving the equations corresponding to a harmonic regime with a complex frequency. Several numerical methods have been tested, to determine some practical rules of convergence. The e!ect of the heavy #uid is pointed out. It is shown that these rules depend on two main parameters: the wavenumber in the plate and the coupling factor.

Then two examples are presented. The "rst one corresponds to a ba%ed plated excited by a turbulent wall pressure. A numerical comparison shows the in#uence of the #uid on the response of the system and the importance of the resonance modes which, obviously, govern the power density spectrum of the system response.

The second example corresponds to a thin shell excited by a transient acoustic wave. The signal di!racted by the shell is expressed as a series of resonance modes and is compared with experimental results obtained in a water tank. The quality of the agreement between numerical predictions and experiments points out that the resonance modes series is certainly one of the best tools to solve this class of problems.

Figure 1 .

 1 Figure 1. Sketch of the #uid-loaded structure.

Figure 3 .

 3 Figure 3. Plate displacement showing convergence of Method I. Plate 2 at 1500 Hz. (;, Nd"30; Np"30, ᭝, Nd"40, Np"40; ᭛, Nd"90, Np"90; continuous line Nd"98, Np"98).

Figure 2 .

 2 Figure 2. Plate displacement showing convergence of Method I. Plate 2 at 56 Hz (;, Nd"12, Np"6; ᭝, Nd"18, Np"20; ᭛, Nd"30, Np"30; continuous line Nd"40, Np"40).

Figure 5 .

 5 Figure 5. Plate displacement showing convergence of Method II. Plate 2 at 1500 Hz. (;, Nd"11; ᭝, Nd"15; ᭛, Nd"19; continuous line, Nd"27).

Figure 4 .

 4 Figure 4. Plate displacement showing convergence of Method II. Plate 2 at 56 Hz. (;, Nd"2; ᭝, Nd"3; ᭛, Nd"4; continuous line, Nd"5).

Figure 6 .

 6 Figure 6. Comparison between methods I and II (marked with ᭛) for plate 2 at 56 Hz.

Figure 7 .

 7 Figure 7. Comparison between methods I and II (marked with ᭛) for plate 2 at 1500 Hz.

Figure 8 .

 8 Figure 8. Directivity pattern at 1 m. Plate 1 (continuous line), plate 2 (marked with ᭛) and plate 3 (marked with ᭝) at 56 Hz.

Figure 9 .

 9 Figure 9. Directivity pattern at 1 m. Plate 1 (continuous line), plate 2 (marked with ᭛) and plate 3 (marked with ᭝) at 1500 Hz.

Figure 10 .

 10 Figure 10. Power spectral density of the velocity of the plate at x"0)9¸V, y"0)9¸W: **, power spectral density of the velocity in water; } --} --, power spectral density of the in vacuo velocity.

Figure 11 .

 11 Figure 11. Geometry of Line-2 and the three co-ordinate systems.

Figure 12 .

 12 Figure 12. Comparison between measurement and prediction of the time-dependent sound "eld di!racted in the axial incidence direction by a Line-2 shell made of aluminum alloy with radius R"0)02 m, total length 4R and thickness h"0)001 m: **, computed signal; } -} -} -, measured signal.

TABLE 1

 1 Characteristics of the three plates

	Material	E			h	h/D	CF
		(Pa)	(kg m\)		(mm) (kg m\ N\) (m\)
	Plate 1: steel	2)26;10	7800	0)28	10	3)82;10\	12)8
	Plate 2: aluminum	0)73;10	2650	0)33	10	3)88;10\	37)7
	Plate 3: aluminum	0)73;10	2650	0)33	3	34)38;10\	125)8

TABLE 3

 3

			TABLE 2
			Method I*values of (Nd!Np) and M
	F (Hz)	56	120	250	250	1500
	Plate 1	6}6; 1	10}10; 1	12}12; 3	30}30; 3	60}60; 7
	Plate 2	30}30; 1	30}30; 2	40}40; 3	40}40; 5	80}80; 8
	Plate 3	30}30; 3	40}40; 5	40}40; 8	60}60; 11	98}98; 18
			Method II*values of (Nd!Np) and M
	F (Hz)	56	120	250	500	1500
	Plate 1	4}8; 1	7}11; 1	8}12; 3	13}17; 3	19}23; 7
	Plate 2	4}8; 1	7}11; 2	9}13; 3	15}19; 5	19}23; 8
	Plate 3	11}15; 3	15}19; 5	19}23; 8	26}30; 11	*

TABLE 4

 4 Resonance frequencies of the plate and peak frequencies of the plate velocity power density spectrum

	Mode no.	Resonance frequencies (Hz)	Peak frequencies
			of S T	(Hz)
	1	2 3 )5 (1!i8)1;10\)	2 3 )6
	2	5 0 )2 (1!i2)0;10\)	5 0 )3
	3	8 1 )9 (1!i6)1;10\)	8 1 )9
	4	9 0 )7 (1!i4)4;10\)	9 0 )6
	5	108)2 (1!i4)6;10\)	108)3
	6	152)3 (1!i2)6;10\)	152)3
	7	152)5 (1!i5)2;10\)	152)9
	8	173)9 (1!i7)5;10\)	176)0
	9	205)3 (1!i2)4;10\)	208)2
	10	216)4 (1!i1)4;10\)	216)4
	11	252)7 (1!i1)0;10\)	254)2
	12	301)2 (1!i3)6;10\)	301)4
	13	314)1 (1!i1)0;10\)	315)0
	14	320)2 (1!i3)7;10\)	320)0
	15	340)4 (1!i1)1;10\)	341)0
	16	346)3 (1!i4)3;10\)	345)0

The authors have been very pleased with the opportunity which has been o!ered them to write a paper for the birthday of their friend Philip Doak. They are sure that, as usual, Phil will have comments and criticisms 2 . They suggest that this discussion takes place when he likes at Les Arcenaux, where "sh and wine are still gorgeous!