2. Initial physical problem#

2.1. Problem description#

We consider a set of identical beams, with axis \(z\), arranged periodically (i.e. \(\varepsilon\) the space period). These beams are located inside an enclosure filled with fluid (see [fig 2.1-a]). We want to characterize the vibratory behavior of such a medium, considering for the time being only the effect of added mass of the fluid, which is preponderant [bib6].

_images/Object_2.svg

Figure 2.1-a

2.2. Modeling hypotheses#

It is considered that the fluid is a perfect fluid that is initially at rest, incompressible. As the hypothesis of small displacements around the equilibrium position has been carried out (fluid initially at rest), the field of movement of the fluid particles is irrotational so that there is a potential for fluid displacement noted \(\Phi\). There is no fluid flow through the outer surface \(\Gamma\).

It is considered that the beams are homogeneous and have a constant cross section according to \(z\in \left]\mathrm{0,}L\right[\). To model the beams, the Euler model is used and the flexural movements are only taken into account. The beam section is rigid and the displacement of any point in the section is noted:

\({\mathrm{s}}^{l}\) the bending of the beam no. \(l\) \(({\mathrm{s}}^{l}(z)\mathrm{=}({\mathrm{s}}_{x}^{l}(z),{\mathrm{s}}_{y}^{l}(z)))\).

The beams are embedded at both ends.

The variational form of the fluid-vibro-acoustic structure problem (conservation of mass, dynamic equation of each tube) is written:

\({\mathrm{\int }}_{{\Omega }_{F}}\mathrm{\nabla }\Phi \mathrm{.}\mathrm{\nabla }{\Phi }^{\text{*}}\mathrm{=}\mathrm{\sum }_{l}{\mathrm{\int }}_{{\gamma }_{l}}({\mathrm{s}}^{l}·\mathrm{n}){\Phi }^{\text{*}}\mathrm{\forall }{\Phi }^{\text{*}}\mathrm{\in }{V}_{\Phi }\) eq 2.2-1

\({\mathrm{\int }}_{0}^{L}{\rho }_{S}S\mathrm{.}\frac{{\mathrm{\partial }}^{2}{\mathrm{s}}^{l}}{\mathrm{\partial }{t}^{2}}\mathrm{\cdot }{\mathrm{s}}^{l\text{*}}+{\mathrm{\int }}_{0}^{L}E\mathrm{I}\mathrm{\cdot }\frac{{\mathrm{\partial }}^{2}{\mathrm{s}}^{l}}{\mathrm{\partial }{z}^{2}}\mathrm{\cdot }\frac{{\mathrm{\partial }}^{2}{\mathrm{s}}^{l\text{*}}}{\mathrm{\partial }{z}^{2}}\text{=}\mathrm{-}{\mathrm{\int }}_{0}^{L}({\mathrm{\int }}_{{\gamma }_{l}}{\rho }_{F}\mathrm{.}\frac{{\mathrm{\partial }}^{2}\Phi }{\mathrm{\partial }{t}^{2}}\mathrm{\cdot }\mathrm{n}){\mathrm{s}}^{l\text{*}}\mathrm{\forall }{\mathrm{s}}^{l\text{*}}\mathrm{\in }{V}_{s}\) eq 2.2-2

with:

\({V}_{s}={({H}_{0}^{2}(]\mathrm{0,}L[))}^{2}\) and \({V}_{\Phi }={H}^{1}({\Omega }_{F})\)

where:

  • \(n\) is the incoming normal to beam \(l\),

  • \({\rho }_{F}\) is the constant fluid density throughout the domain,

  • \({\mathrm{\rho }}_{S}\) is the density of the material constituting the beam,

  • \(S\) is the section of the beam,

  • \(E\) is Young’s modulus,

  • \(\mathrm{I}\) is the inertia tensor for the section of the beam.

The second member of equation [éq 2.2-2] represents the forces exerted on the beam by the fluid. Fluid pressure \(p\) is linked to displacement potential by: \(p\mathrm{=}\mathrm{-}{\rho }_{F}\frac{{\mathrm{\partial }}^{2}\Phi }{\mathrm{\partial }{t}^{2}}\). In the same way, the second member of equation [éq 2.2-1] represents the fluid flow induced by the movements of the beams. At the border of each \(l\) beam we have: \({\mathrm{s}}^{l}\mathrm{\cdot }\mathrm{n}\mathrm{=}\mathrm{\nabla }\Phi \mathrm{\cdot }\mathrm{n}\).

This formulation leads to a non-symmetric matrix system, which is not very convenient, in

especially when looking for vibration modes.