SEA APPROACH OF THE HIGH FREQUENCY DYNAMIC ANALYSIS OF BOUNDED STRUCTURES COUPLED WITH HEAVY COMPRESSIBLE UNBOUNDED FLUIDS

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SEA APPROACH OF THE HIGH FREQUENCY DYNAMIC ANALYSIS OF BOUNDED STRUCTURES

COUPLED WITH HEAVY COMPRESSIBLE UNBOUNDED FLUIDS

F. Chabas

To cite this version:

F. Chabas. SEA APPROACH OF THE HIGH FREQUENCY DYNAMIC ANALYSIS OF BOUNDED

STRUCTURES COUPLED WITH HEAVY COMPRESSIBLE UNBOUNDED FLUIDS. Journal de

Physique Colloques, 1990, 51 (C2), pp.C2-981-C2-984. �10.1051/jphyscol:19902229�. �jpa-00230556�

ler Congress Frangats d'Acoustique 1990

SEA APPROACH OF THE HIGH FREQUENCY DYNAMIC ANALYSIS OF BOUNDED STRUCTURES COUPLED WITH HEAVY COMPRESSIBLE UNBOUNDED FLUIDS

F. CHABAS

Office National d'Etudes et de Recherches A6rospatiales, BP. 72, F-92322 Ch&tlllon Cedex, France

Résumé - Dans le cadre de l'étude dynamique linéaire hautes fréquences des structures sous-marines, on présente une nouvelle formulation de l'Analyse Statistique Energétique. On propose dans un premier temps une modification des équations de base de la SEA pour les structures couplées avec des fluides denses non bornés. On présente dans un deuxième temps des méthodes de calcul des paramètres SEA (facteurs de perte par couplage) qui permettent d'affaiblir les hypothèses de base de la méthode. La troisième partie est consacrée aux applications et comparaisons expérimentales.

Summary - In the framework of the high frequency linear dynamic analysis of underwater structures, we present a new formulation of Statistical Energy Analysis. In a first time, we propose a modification of basic equations of SEA for structures which are coupled with heavy unbounded fluids. Then, we present some methods to determine the SEA parameters (coupling loss factors) which allow to weaken the basic assumptions of the method. The third part is devoted to numerical applications and experimental comparisons.

1-INTRODUCTION

Owing to the simplicity of its formalism, Statistical Energy Analysis (SEA) provides a very easily used and very powerful tool for the high frequency linear dynamic analysis of complex mechanical systems. In counterpart, in this kind of method, the whole of the difficulty lies in the determination of SEA parameters of mechanical systems, and specially coupling loss factors. These difficulties are related to:

- the basic assumptions of SEA which do not allow, for instance, to take into account strong couplings.

- the methods used in the determination of coupling loss factors. These methods are generally based on models of energy transmission between semi-infinite media which do not always correspond to the real phenomena.

- the basic formulation of SEA (l) which does not allow to deal with structures which are coupled with heavy unbounded fluids.

In this context, it seems necessary to develop a new formulation of SEA in order to extend it capabilities and to improve the quality of its numerical predictions.

2 - NEW FORMULATION FOR HYDRO-ELASTIC COUPLINGS

When a structure is coupled with a light unbounded fluid, the fluid effects are only of dissipative type (radiation to infinity). The definition of the total mechanical energy of the coupled system is that of the structure in vacuum and fluid effects are introduced through a fluid dissipation loss factor n.H(wo)> which is related to a radiation efficiency coefficient om<i(coo) such that:

(1) o > ) = ° SH °

rad ° PFCFS

where Mg is the structural mass, S the radiating area, PF Cjr the acoustical fluid impedance and co⁰ the central frequency of the analysed band.

For a structure coupled with an heavy unbounded fluid, it is necessary to take into account the added mass effects of the fluid. In order to preserve the power balance aspect of SEA equations, we are led to introduce a new parameter pjj (w0), called "equivalent added mass", which is frequency dependent and which allows to define the mean total energy of the coupled system in the form:

(2) <Etot> = [p(m) +pH(a>o))<U(m,t).U{m.,t)>do(m)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19902229

C2-982 COLLOQUE DE PHYSIQUE

where D is the domain occupied by the structure, p(m) the mass density relatively to ddrn), ~ ( r n , t ) the velocity field in the elastic medium and where

<.>

represents averaged quantities on the analysed band.

For such a system, and to remain coherent, we will define the radiation efficiency by:

3 -SEA PARAMETERS FOR ISOLATED SYSTEMS

For a bounded elastic media, the equation of vibrations can be written, in the frequency domain:

where M, C, and K are respectively the mass, damping and stiffness matrix of the system.

Equation (4) is assumed to be discretized on the eigenmodes of the associated conservative medium.

If the medium is coupled with an heavy unbounded fluid, we include in M(o) and C (a) the added mass and dissipation matrix of the fluid.

The resonant frequencies are then the solutions0 u = of equation:

If there is no external fluid or if the fluid is a light one, the matrix M is frequency independant.

Moreover, equations (5) are uncoupled so that the solution can be immediately obtained.

On the contrary, if there is an heavy unbounded fluid, equations (5) are coupled. In the high frequency range, we can reasonably neglect the intermodal couplings. Nevertheless, the equations must be solved numerically because the matrix M is frequency dependant in this case.

By a simple counting process, we can then determine, from the knowledge of the resonant frequencies, the mean modal densities of the system.

In order to calculate the other SEA parameters, we use the exact power balance equations of the system. For a single fluid-structure coupled system, SEA power balance gives:

We suppose the structure is excited by a time-stationary, spatially delta-correlated random field, defined by its power spectral density SF(^). Relation (4) then shows that the stationary response of the system is given by :

(7) ^{Sv (a) =} H ( o ) S d o ) H* (a)

with H(o) = [- w2 M(w)

+

i w C(w)

+

^{a-1 and H*(o) =}

fiT(o)

The actual mean total energy of the system is:

(8) <Etot> =

- 1

t r { o 2 ~ ( o ) ~ , ( w ) } d o + ;

'1

t r { K S v ( w ) ) d o

2 Bug B U B

From an SEA standpoint, this energy is written (assuming an homogeneous elastic medium):

and the mean input power i s defined by:

4 -DETERMINATION OF COUPLING LOSS FACTORS For two coupled systems, the SEA power balance equation gives:

Nl(41 + 912) - Nz 421

[I?l 412

-

N2

+

^{4 . 4} =

[ ~ ~ ~ : ~ ~ ]

with N1, N2 the mean modal densities, q l , q2, the mean dissipation loss factors (which are assumed to be known) and q12,921, the unknown coupling loss factors.

Equation (11) can be rewritten as:

112)

We can calculate the columns of matrix A by successively exciting each of the systems with a stationary, delta-correlated random field and by determining the total mean energies of the systems.

The com utation i s based on a particular technique (2) in which the dynamics of each system is describefby mean of two fields:

-

one field which i s related to the internal degrees of freedom and discretized on the modal basis of the system

-

one field which describes the coupling between the systems and which is discretized on a n arbitrary functional basis of the boundary.

Such an approach allows to describe the actual phenomena in energy transmission and can take into account the wave conversions in the jonctions.

By inverting the matrix A, we can determine the coupling loss factors 1112 and q2l and check the method by reidentifying the dissipation loss factors q1 and q2.

Numerical applications have been made in the case of rectangular plates, circular plates and cylindrical shells. For the calculation of the hydrodynamic operators, the elastic medium is assumed to be set in an infinite rigid baffle.

For structures coupled with heavy unbounded fluids, the results (figures 1, 2, 3) have permitted to enhance new phenomena near the coincidence frequency.

For coupling between systems, the results also show (figure 4) t h a t coupling loss factors are conditioned by the internal loss factors of the systems.

Experimental comparisons and calculation of the radiated pressure field have also been presented in the oral conference.

We have proposed some methods to improve the numerical prediction of SEA. These methods do not lead to reconsidering the power balance aspect of the equations. Moreover, they are rigourous from a mechanical standpoint and allow hereby to take into account strong coupling in the SEA formalism.

REFERENCES

111 LYON, R.H., "Statistical Ener y Analysis of Dynamic Systems: Theory and Application", Cambridge, Massachussets and fondon, England: the MIT Press (19'75).

/2/ CHABAS, F., "Approche SEA des couplages hydro-Clastiques HF

-

DBtermination des facteurs de perte par couplage", RT ONERA no 8813454 RY 088 (1989).

C2-984 COLLOQUE DE PHYSIQUE

! , , , , , , , , , , , , , . t ~ , FIFC I

0 7.5 10-1 1.5 2.3 3

Shell in vacuum ^{. -}

-

Shell in water

-

Fig. 1. Modal densities of a cylindrical shell Fig. 2.Normalized added mass of a submerged FC = coincidence frequency. cylindrical shell.

1

^WFC

0 ' 7.5 10-1 1.5 2.3 Fig. 4.Coupling loss factor between a cylindri-

cal shell and a circular transverse plate:

<c = critical damping o f the shell Fig. 3.Radiation efficiency of a submerged

Sp

= 0.005 (critical damping o f the

cylindrical shell. plate).