1 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 Métiers (CNAM) Chaire de Mécanique Laboratoire de Mécanique des structures et des systèmes couplés GDR IFS/jeudi 26 juin 2008/14h - 16h

2 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

3 Vibrations of elastic structures

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10 FREQUENCY DOMAIN Measured Transfer Function

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20 REDUCED ORDER MODEL

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23 Non-homogeneous heavy compressible fluid Plane irrotationality

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26 STRUCTURAL ACOUSTIC VIBRATIONS

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37 STRUCTURAL ACOUSTICS EQUATIONS Structure submitted to a fluid pressure loading

38 Structure submitted to a fluid pressure loading Mechanical elastic stiffness contribution Geometric stiffness contribution Rotation of the normal contribution

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51 REDUCED ORDER MODEL Dynamic substructuring decomposition (Craig-Bampton-Hurty) Fixed/free eigenmodes Static interface deformations

52 Fluid submitted to a wall normal displacement Linearized Euler equation Constitutive equation Kinematic boundary condition For, we impose in on in

53 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

54 Local fluid equations in terms of pressure and wall normal displacement with the constraint in on Helmholtz equation Kinematic boundary condition

55 Introduction of the displacement potential field with the uniqueness condition

56 Local fluid equations in terms of displacement potential field and wall normal displacement on in with the uniqueness condition

57 Pressure-Displacement Unsymmetric Variational Formulation with the constraint

58 Reduced Order Model First basic problem Acoustic modes in a rigid motionless cavity Orthogonality conditions:

59 Reduced Order Matrix Model Second basic problem The static pressure solution

60 Symmetric Matrix Reduced Order Model Decomposition of the admissible space into a direct sum of admissible subspaces: Solution searched under the following form: p and u.n satisfy the constraint

61 Symmetric Matrix Reduced Order Model where represents a “pneumatic” operator (quasistatic effect of the internal compressible fluid)

62 Symmetric Matrix Reduced Order Model Hybrid FE/generalized coordinates representation with

63 Symmetric Matrix Reduced Order Model Generalized coordinates representation

64 HEAVY / LIGHT COMPRESSIBLE FLUID Gas 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)

65 COMPUTATION-EXPERIMENT COMPARISON

66 Kelvin-Voigt model Thin interface dissipative constitutive equation Structural-acoustic problem with interface damping Particular linear viscoelastic constitutive equation (cf Ohayon-Soize, Structural Acoustics and Vibrations, Academic Press, 1998)

67 absorbing material Interface wall damping equation Interface wall damping impedance effects

68 Boundary value problem in terms of (u,  p,  ) Symmetric formulation of the spectral structural-acoustic problem

69 CONCLUSION – OPEN PROBLEMS Appropriate Reduced Order Models for Broadband Frequency Domains Hybrid Passive / Active Treatments for Vibrations and Noise Reduction Nonlinearities (Vibrations / Transient Impacts and Shocks)