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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 base Morand-Ohayon /Fluid Structure Interaction / Wiley 1995 Conservatoire National des Arts et M é tiers (CNAM) - PowerPoint PPT Presentation
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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)[email protected]
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 c o u s tic
co m p re s s 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)
