Compressible fluid-structure interaction and modal representation

Vibrations of structures in civil engineering, aerospace, biomechanics are frequently connected with fluid influence. Fluid is a part of the mechanical system, and compressible gas or liquid. It is the significant component of the whole mechanical model. Four different dam water reservoir models, the first rigid dam incompressible water, the fourth flexible dam compressible water, are presented by Tiliouine, Seghir [1]. Galerkin variational formulation is established for each model and earthquake response studies presented. A method to compute the vibration modes of an elastic shell or plate in contact with a compressible fluid is considered by Hernandez [2]. Presence of zero-frequency spurious modes with no physical meaning is indicated. Elastoacoustic vibration modes are investigated by Mellado and Rodriguez [3]. Interaction of compressible flow and deformable structures is solved by Gretarsson et al. [4]. Hydrodynamic pressure on underwater glide vehicle and surface stresses are investigated by Du et al. [5]. Vibrations in magnetorheological fluids are studied by Bansevičius et al. [6]. Forced vibrations of two plates in incompressible fluid are investigated in [7]. These two plates, not connected together, interact through an incompressible fluid. Interaction of the different eigenmodes of the same plate in vacuum is also presented.


Introduction
Vibrations of structures in civil engineering, aerospace, biomechanics are frequently connected with fluid influence.Fluid is a part of the mechanical system, and compressible gas or liquid.It is the significant component of the whole mechanical model.Four different dam -water reservoir models, the first rigid dam -incompressible water, the fourth flexible dam -compressible water, are presented by Tiliouine, Seghir [1].Galerkin variational formulation is established for each model and earthquake response studies presented.A method to compute the vibration modes of an elastic shell or plate in contact with a compressible fluid is considered by Hernandez [2].Presence of zero-frequency spurious modes with no physical meaning is indicated.Elastoacoustic vibration modes are investigated by Mellado and Rodriguez [3].Interaction of compressible flow and deformable structures is solved by Gretarsson et al. [4].Hydrodynamic pressure on underwater glide vehicle and surface stresses are investigated by Du et al. [5].Vibrations in magnetorheological fluids are studied by Bansevičius et al. [6].
Forced vibrations of two plates in incompressible fluid are investigated in [7].These two plates, not connected together, interact through an incompressible fluid.Interaction of the different eigenmodes of the same plate in vacuum is also presented.

Equations of plate motion
Deflections of a plate AB (Fig. 1), supported at opposite edges, can be approximated by the functions of distance y and time functions   where n is any integer.The base functions

 
or 0 where are orthogonal, the matrix D and may be the matrix C is diagonal.
When the fluid is compressible and inviscid, the classical Helmholtz equation speed in the fluid.By using the separation of variables method the velocity potential can be expressed   , solution coincides with [7].
The boundary condition on the line where rs H   , and d is width of the plate, parallel to the axis z, perpendicular to the x, y plane.

Eigen frequencies and modal representation
When vibrations are harmonic

 
. Density of the air is much less then the density of water, and diminution of frequency in the fluid is not so significant (

Discussion
When vibrations are forced by harmonic force ... n  is acceptable.The set of eigenmodes can be unsuitable as base functions because the set can be not complete in the vector space of investigation.Moreover, the eigenmodes are not orthogonal.It may be indicated, that added masses are useful only when rigid bodies are in fluid.In some sense the coefficients rs sr

 
can be presented as substitute to the added mass.
The eigenmodes are important when resonant vibrations are induced and one or two eigenvectors of the corresponding matrix are required to present the forced vibration.Real fluid always is compressible, so any investigation of the fluid and structure raise the problemwhat is the practical and general theoretic significance of the fluid compressibility.


holds true, where o c is the sound

Particular solution ( 3 )
depends on the frequency ω = 2πf, and this changes the whole solution of the fluidstructure interaction problem.If the sound speed   cides or is near the eigenfrequency j f of the plate in com- pressible fluid (underlined values in Tables1, 2), the mode of vibration can be assumed equal to the eigenvectorthe j-th column in the matrix mode of the forced vibrations should be more precise, the other eigenvectors of the matrix   Hs Tf can be applied.If all n eigenvectors are necessary, any set of eigenvectors in the line with θ o = 0 of the Table1.But the first eigenvalue of the matrix H B  depend on the vibration frequency, the eigenval- ues s  and eigenvectors    were performed, the eigenvalues in the lower lines s = 2,3,…,7 of theTable1.Every eigenvalue s f of the corresponding matrix   Hs Bf is almost the same as o .

Table 2
MODE T.