Keywords: vibrations, medium-frequency range, complex rays, heterogeneous structures, uncertainties.

The modeling and resolution of medium-frequency vibration problems is a major challenge in the design of satellites or car chassis. The difficulty lies in the very small length of variation of the phenomena being studied compared to the structure's characteristic dimension [1]. Different approaches have been proposed, such as improved finite elements [2], specific reduced bases [3] or SEA extensions [4], but most of these require particular geometries or additional information in order to give predictive results.

The Variational Theory of Complex Rays (VTCR) [5] is a predictive tool specifically intended for medium-frequency problems. Initially, this theory was based on the assumption that the structure is an assembly of homogeneous substructures. Each substructure is associated with a superelement described by degrees of freedom which correspond to local basic modes defined on two scales and satisfying the dynamic equations. This strategy has been validated for three-dimensional assemblies of elastic homogeneous plates with low damping. An extension of the VTCR [6] enables the designer to take into account structural discontinuities, whether intentional (such as portholes or equipment connections) or unintentional (such as cracks). Each resulting heterogeneous substructure and corresponding superelement includes a family of additional basic modes, weighted in order to verify a priori conditions on the boundary of the discontinuity. The weights are calculated and stored in advance for ranges of parameters corresponding to each type of heterogeneity. Thus, the evaluation of effective quantities (deformation energy, vibration level,...) requires only the resolution of a small constrained system of equations.

In this paper, improvements are proposed to deal with sensitivity analysis. The purpose is to evaluate the influence on the structural response of discontinuities whose positions, sizes or directions are uncertain. A specific partial inversion of the constrained system of equations enables one to carry out multiple resolutions inexpensively for a range of parameters defining the possible shapes of the discontinuity.

1

A. Deraemaeker, I. Babuska, P. Bouillard, "Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions", Int. J. Numer. Meth. Engng. 46, pp. 471-499, 1999. doi:10.1002/(SICI)1097-0207(19991010)46:4<471::AID-NME684>3.0.CO;2-6

2

A. A. Oberai, P. M. Pinsky, "A multiscale finite element method for the Helmholtz equation", Comput. Methods Appl. Mech. Engrg., 154, pp. 281-297, 1998. doi:10.1016/S0045-7825(97)00130-8

3

C. Soize, "Reduced models in the medium frequency range for general dissipative structural-dynamics systems", Eur. J. Mech., A/Solids, 17/4, pp. 657-685, 1998. doi:10.1016/S0997-7538(99)80027-8

4

M. N. Ichchou, A. Le Bot, L. Jézéquel, "Energy models of one dimensional multi-propagative systems", Journal of Sound and Vibration, 201, pp. 535-554, 1997. doi:10.1006/jsvi.1996.0780

5

P. Ladevèze, L. Arnaud, P. Rouch, C. Blanzé, "The variational theory of complex rays for the calculation of medium-frequency vibrations", Engineering Computations, Vol. 18 No.1/2, pp. 193-214, 2001. doi:10.1108/02644400110365879

6

L. Blanc, C. Blanzé, P. Ladevèze, P. Rouch, "A multiscale and "Trefftz" computational method for medium-frequency vibrations of assemblies of heterogeneous plates", Computer Assisted Mechanics and Engineering Sciences, vol. 10, pp. 375-384, 2003.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)