On the vibration problem for an elastic body surrounded by a slightly compressible fluid

The vibration problem for au plastic -bmmdeditody immersed itu&n; idealfktid small compressibility (associated with the small parameter s ~> 0) is considered in the framework of scattering theory. ït is shown that the scattering frequencies converge as e -* 0 to the eigenfrequencies of the self adjoint problem associated with an incompressible fluid. The asymptotic behaviour of the eigenfrequencies for small e is given via the implicit function theorem. Some gênerai considérations about the physics of the problem and expérimental results are also given. Résumé. — NOMS considérons le problème de vibration d'un corps élastique borné plongé dans un fluide idéal de petite compressibilité (associé au petit paramètre e -* 0) dans le cadre de la théorie de la diffusion (scattering). Nous montrons que les fréquences de scattering convergent lorsque s ^ 0 vers les fréquences propres du problème autoadjoint correspondant à un fluide incompressible. Le comportement asymptotique des fréquences propres pour e petit est obtenu via le théorème de fonctions implicites. Nous donnons aussi certains résultats expérimentaux et des considérations générales sur le problème physique. I. GENERALITIES ABOUT THE PHYSICAL PROBLEM It is proposée to investigate the problem of the low-frequency forced vibrations of a structure immersed in an infinité liquid medium. The main response parameters are : — the displacement field at any point of the structure, — the pressure field in the neighbourhood of the structure (the so-called near-field pressure), — the far-field pressure, (*) Received on May 1982. (*) Office National d'Études et de Recherches Aérospatiales, 92320 Châtillon. () Laboratoire de Mécanique Théorique, Tour 66, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75005 Paris. R.A.I.R.O. Analyse numérique/Numerical Analysis, 0399-0516/1983/311/K 5.00 © AFCET-Bordas-Dunod 312 R. OHAYON, E. SANCHEZ-PALENCIA as functions of the excitation frequency (considérée as a continuously increasing parameter). It should be emphasized that the expérimental analysis follows the above scheme by using displacement and pressure déviées, the excitation frequency parameter being controlled by the experimentalist through appropriate devices. If L (resp. A) dénotes a characteristic length of the structure (resp, the wavelength of the radiated wave), the low-frequency range considered here corresponds to the case A |> L. Expérimental analysis carried on test specimens of révolution immersed in an " infinité " liquid medium with a free surface — for example, ring-stiffened cylinders — have shown that, for low-frequency vibration, the response curves present " resonant peaks " similar to classical slightly damped conservative linear Systems analysed theoretically through asymptotic expansions [2], Moreover, the values of the frequencies corresponding to those peaks were " practically " the same for the three field parameters described above. Therefore, a new set of response parameters is : — the " resonant frequencies " corresponding to the successive peaks of the response spectrum and the corresponding structural displacement field and liquid nearand far-field pressure. Let us now discuss the origin of the damping of the fluid-structure coupled system-. Refering again to the expérimental analysis, for example for a test specimen of 1 m height and 03 m of radius, for a range of low=frequency between 40 Hz and 120 Hz, the damping effect s arising either from the radiation of gravity waves or from the viscosity of the fluid were negligible in comparison with the two other sources of damping, namely the one occuring in the structure dissipation and the one arising from the compressibility of the liquid (radiation). It should be recalled that for bounded fluid-structure Systems, the System is conservative even if the fluid is compressible [8, 10], which is not the case if the external fluid volume is infinité as a conséquence of the radiation phenomena. It results from the above expérimental considérations and from theoretical and numerical investigations (without a free surface in [2], and with a free surface in [9]), that the conservative fluid-structure System associated with the real physical System corresponds to the following hypotheses : — elastic structure, — incompressible liquid, — free surface with a Dirichlet boundary condition dénotes the displacement potential of the fluid). The last condition will not be considered in the present paper. Expérimental R.A.I.R.O. Analyse numérique/Numerieal Analysis VIBRATION PROBLEM FOR AN ELASTIC BODY 313 investigations bring to the conclusion that — for instance, in the case of a vertical cylinder immersed in a liquid at a 1 meter depth — the results differ slightly from the case where the specimen is completely surrounded by the fluid filling the whole space. We have already said that the case considered hère corresponds to A > L. By considering the Helmholtz équation in the fluid, it is easy to see that the behaviour of the liquid in the neighbourhood of the structure is governed by the Laplace équation Acp — 0 (incompressible limit). This fact is verified by the expérimental analysis (for the low-frequency behaviour) as very sharp peaks are observed concerning the structural displacement and the very near pressure field.