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Formulations variationnelles et modèles réduits pour les vibrations de structures contenant des fluides compressibles en l’absence de gravité

Cours de Roger Ohayon

référence de baseMorand-Ohayon /Fluid Structure Interaction / Wiley 1995

Conservatoire National des Arts et Métiers (CNAM)Chaire de Mécanique

Laboratoire de Mécanique des structures et des systèmes couplés (www.cnam.fr/lmssc)ohayon@cnam.fr

GDR IFS/jeudi 26 juin 2008/14h - 16h

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Some local references

H. Morand, R. Ohayon / Fluid Structure Interaction/ Wiley – 1995 (chap. 1, 2, 7, 8, 9)

R. Ohayon, C. Soize / Structural Acoustic and Vibrations,Academic Press, 1998

R. Ohayon / Fluid Structure Interaction problems / Encyclopedia of Computational Mechanics, vol. 2, Chap. 21, Wiley, 2004

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Vibrations of elastic structures

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FREQUENCY DOMAIN

Measured Transfer Function

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REDUCED ORDER MODEL

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Non-homogeneous heavy compressible fluid

C h a rt T it le

s lo sh ing

In co m p ress ib leh o m og en eo us

a co u s tic

co m p re ss ib leh o m og en eo us

in te rn a l g ra v ityw a ves

co m p re ss ib len o n-h om o g en eo us

D yn a m ic o f liq u ids

Plane irrotationality

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STRUCTURAL ACOUSTIC VIBRATIONS

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STRUCTURAL ACOUSTICS EQUATIONSStructure submitted to a fluid pressure loading

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Structure submitted to a fluid pressure loading

Mechanical elastic stiffness contribution

Geometric stiffness contribution

Rotation of the normal contribution

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REDUCED ORDER MODEL

Dynamic substructuring decomposition (Craig-Bampton-Hurty)

• Fixed/free eigenmodes• Static interface deformations

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Fluid submitted to a wall normal displacement

Linearized Euler equation

Constitutive equation

Kinematic boundary condition

For , we impose

in

in

on

in

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Static pressure field and normal wall displacement relation

This case corresponds to a zero frequency situation

in which denotes the measure of the volume occupied by domain

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Local fluid equations in terms of pressure and wall normal displacement

with the constraint

in

on

Helmholtz equation

Kinematic boundary condition

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Introduction of the displacement potential field

with the uniqueness condition

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Local fluid equations in terms of displacement potential field and wall normal displacement

on

in

with the uniqueness condition

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Pressure-Displacement Unsymmetric Variational Formulation

with the constraint

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Reduced Order Model

First basic problem Acoustic modes in a rigid motionless cavity

Orthogonality conditions:

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Reduced Order Matrix Model

Second basic problem The static pressure solution

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Symmetric Matrix Reduced Order Model

Decomposition of the admissible space into a direct sumof admissible subspaces:

Solution searched under the following form:

p and u.n satisfy the constraint

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Symmetric Matrix Reduced Order Model

where

represents a “pneumatic” operator (quasistatic effect of the internal compressible fluid)

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Symmetric Matrix Reduced Order Model

Hybrid FE/generalized coordinates representation

with

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Symmetric Matrix Reduced Order Model

Generalized coordinates representation

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HEAVY / LIGHT COMPRESSIBLE FLUIDGas or Liquid (with/without free surface)

The SYMMETRIC reduced order matrix models should be employed with

great care:

For a light fluid, structural in-vacuo modes can be used

For a heavy fluid – liquid, structural in-vacuo modes lead to poor convergence and MANDATORY, added-mass effects must be introduced (this now classical aspect can be introduced via a quasi-static so-called correction or, which is exactly the same, via an added mass operator provided the starting variational formulation contains a proper basic static behavior)

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COMPUTATION-EXPERIMENT COMPARISON

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Kelvin-Voigt model

Thin interfacedissipative

constitutive equation

Structural-acoustic problemwith interface damping

Particular linear viscoelastic constitutive equation (cf Ohayon-Soize, Structural Acoustics and Vibrations, Academic Press, 1998)

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absorbing material

Interface wall damping equation

Interface wall damping impedance effects

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Boundary value problem in terms of (u, p, )

Symmetric formulationof the spectral structural-acoustic problem

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CONCLUSION – OPEN PROBLEMS

http://www.cnam.fr/lmssc

Appropriate Reduced Order Models for Broadband Frequency Domains

Hybrid Passive / Active Treatments for Vibrations and Noise Reduction

Nonlinearities (Vibrations / Transient Impacts and Shocks)