La Theorie Variationnelle des Rayons Complexes pour le calcul des vibrations moyennes frequences
Keywords:
vibrations, medium frequency, plate, complex raysAbstract
A new approach named the "Variational Theory of Complex Rays" is introduced for computing the vibrations of elastic structures weakly damped in the medium frequency range. Emphasis has been placed here on the most fundamental aspects. The effective quantities (elastic energy, vibration intensity ... ) are evaluated after computing a small system of equations which does not derive from a finite element dicretization of the structure. Numerical examples related to plates show the interest and the possibilities of the VTRC.
Downloads
References
[BAB 95] BABUSKA I., IHLENBURG F., PAIKAND E. et SAUTER S., «A generalized finite
element method for solving the Helmotz equation in the two dimensions with minimal
pollution», Comput. Methods Appl. Mech. Engrg, vol. 128, p. 325-359, 1995.
[BAR 98] BARBONE P., MONTGOMERY J., MICHAEL 0. et HARARI 0., «Scattering by an
hybrid asymptotic/finite element method», Comp. Methods Appl. Mech. Engrg, vol. 164,
p. 141-146, 1998.
[BEL 75] BELOV V. et RYBACK S., « Applicability of the transport equation in the one dimensional
wave propagation problem>>, Akust. Zh., vol. 21, p. 173-180, 1975.
[BEL 77] BELOV V. et RYBACK S., «Propagation of vibrational energy in absorbing structures
>>, Journal of Soviet Physics Acoustics, vol. 23/2, p. 115-119, 1977.
[BOU 94] BOUCHE D. et MOLINET F., Methodes asymptotiques en electromagnetisme. Coli.
Mathematiques et applications, Springer, 1994.
[BOU 98] BOUILLARD P. et IHLENBURG F., Error estimation and adaptativity for finite element
solution in acoustics in Advances in Adaptative Computational Methods in Mechanics.
Ladeveze P., Oden J.T. (eds), Elsevier, 1998.
[BUY 80] BUVAILO L. et IONOV A., << Application of the finite element method to the investigation
of the vibroacoustical characterisics of structures at high audio frequencies >>,
Journal of the Soviet Physics Acoustics, vol. 26 (4), p. 277-279, 1980.
[CUS 90] CUSCHERI J., << Vibration transmission through periodic structures using a mobility
power flow approach >>, Journal of Sound and vibration, vol. 143 (1 ), p. 65-74, 1990.
[OEM 92] DEMKOwiCZ L., KARAFIAT A. et 0DEN J., << Solution of elastic scattering problems
in linear acoustics using h-p boundary element method >>, Comput. Methods Appl.
Mech. Eng., vol. 101, p. 251-282, 1992.
[DER 99] DERAEMAEKER A., BABUSKA I. et BOUILLARD P., <
the FEM solution for the Helmholtz equation in one, two and three dimensions >>, Int. J.
Numer. Meth. Engng., vol. 46, p. 471-499, 1999.
[DOW 85] DowELL E. et KUBOTA Y., <
dynamical systems>>, Journal of Applied Mechanics, vol. 52, p. 949-957, 1985.
[GIR 93] GIRARD A. et DEFOSSE H., <
analysis: application to beam trusses >>, Journal of Sound and Vibration, vol. 65 (I), p.
-170, 1993.
[GRE 99] GREENSTADT J., << Solution of wave propagation problems by the cell discretisation
method>>, Comput. Methods Appl. Mech. Engrg., vol. 174, p. 1-21, 1999.
[GRO 98] GROSH K. et PINSKY P., << Galerkin generalized least square finite element methods
for time harmonic structural acoustics>>, Comp. Methods Appl. Mech. Engrg., vol. 154, p.
-318, 1998.
[HAR 96] HARARI I. et AAND J.R. HUGES K. G.,<< Recent developments in finite element
methods for structural acoustics >>, Arch. of Comp. Meth. Eng., vol. 3, p. 131-311, 1996.
[HOC 93] HOCHARD C., LADEVEZE P. et PROSLIER L., << A simplified analysis of elastic
structures>>, Eur. J. Mech. A/Solids, vol. 12(4), p. 509-535, 1993.
[ICH 97] ICHCHOU M., BOT A. L. et JEZEQUEL L., << Energy model of one-dimensional,
multipropagative systems >>, Journal of Sound and Vibration, vol. 201 (5), p. 535-554,
[ILH 95] ILHENBURG F. et BABUSKA I.,<< Dispersion analysis and error estimation of Galerkin
finite element methods for Helmotz equation>>, Int. Journ. Num. Meth. Eng., vol. 38,
p. 3745-3774, 1995.
[ILH 97] lLHENBURG F. et BABUSKA I., << Finite element solution of th Helmotz equation
with high wave number Part 2: the h-p version of the F.E.M. >>, SIAM Num. Anal., vol. 34
(1), p. 315-358, 1997.
[LAD 83] LADEVEZE P., R.D.M. tridimensionnelle, Rapport Interne N·12 L.M.T. Cachan,
[LAD 95] LADEVEZE P., Prediction des vibrations moyennes frequences: Etat de l'art et remarques,
Rapport technique, NT Aerospatiale YX/SA 116 471, 1995.
[LAD 96a] LADEVEZE P., << A new computational approach for structure vibrations in the
medium frequency range>>, C. R. Acad. Sci. Paris., vol. t.322, Serie II b, n· 12, p. 849-856,
[LAD 96b] LADEVEZE P., Une nouvelle approche pour le calcul des vibrations moyennes frequences,
Rapport technique, NT Aerospatiale YX/SA 119 639, 1996.
[LAN 91] LANGRE E. D., Fonctions de transfert de plaques en flexion par equations integrales.
Test de validation et de performance, Rapport technique, CEA :DMT/90/395, 1991.
[LAS 94] LASE Y., ICHCHOU M. et JEZEQUEL L., «Energy flow analysis of bars and beams:
theoretical formulation», Journal of Sound and Vibration, vol. 192 (I), p. 2981-3005, 1994.
[LEU 62] LEUNG A. et CHAN J ., « Fourrier p-element for the analysis of beams and plates >>,
JASA, vol. 34 (5), p. 623-639, 1962.
[LIU 91] LIU W., ZHANG Y. et RAMIREZ M., «Multiple scale finite element methods>>, Int.
J. Num. Meth. Engng, vol. 32, p. 969-990, 1991.
[LUZ 94] LUZZATO E., EDF Workshop proceeding on methods in medium and high frequency:
the alternative to S.E.A, 1994.
[LYO 67] LYON R. et MAIDANICK G.,<< Power flow between linearly coupled oscillators >>,
JASA, vol. 34 (5), p. 623-639, 1967.
[MAC 94] MACE B., <
and some implications of its failure >>, JASA, vol. 178 (I), p. 95-112, 1994.
[MOR 92] MORAND J., << A modal hybridization method for the reduction of dynamic models
>>, In New Advances Computational Structural Mechanics, p. 347-365. Ladeveze P.,
Zienkiewicz O.C. (eds), Elsevier, 1992.
[NEF 89] NEFSKE D. et SUNG S., <
basic theory and application of beams >>, Journal of Vibration, Acoustic, Stress and Reliability
in Design, ASME, vol. Ill, p. 94-100, 1989.
[OHA 89] OHAYON R., << Local and global effects in the vibration of structures. A review
Synthesis>>, In ESA Workshop Proceeding on modal representation of flexible structures
by continuum methods, Noordwijk (Netherlands), p. 29-54, 1989.
[RIZ 85] RIZZO F., SHIPPY D. et REZAYAT M., << A boudary integral equation method for
radiation and scattering of elasic waves in the three dimensions >>, Int. J. Num. Meth. Eng.,
vol. 21, p. 115-129, 1985.
[ROS 97] ROSENHOUSE G., AYRASHI J. et MICHAEL 0., <
boundary spectral line strips>>, Engineering computations, vol. 15 (2), p. 221-232, 1997.
[SOl 85] SOIZE C.,<< The local effects in the linear dynamic analysis of structures in the medium
frequency range >>, In Local Effects in the Analysis of Structures, p. 253-278. Ladeveze
P. (ed), Elsevier, 1985.
[SOl 98] SOIZE C., <
structural-dynamics systems>>, European Journal of Mechanics A/Solids, vol. 17 (4), 1998.
[WIB 96] WIBERG N., BAUSYS R. et HAGER P., << Improved eigen frequencies and eigenmodes
in free vibration analysis », In Advances in Finite Element Technology, p. 43-54.
Topping B.H.V. (ed) Civil Comp. Press, 1996.
[WU 98] Wu K. et GINSBERG J., <
bodies using the surface variational principe >>,Journal of Vibration and Acoustics, vol.
, p. 392-400, 1998.
[ZIE 87] ZIELINSKI A. et HERRERA I., « Trefftz method: Fitting boudary conditions >>, Int.
J. Num. Meth. Eng., vol. 24, p. 871-891, 1987.