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SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE FORMULATION AND MODAL ANALYSIS

SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE FORMULATION AND MODAL ANALYSIS. Ahlem ALIA presented by Nicolas AQUELET Laboratoire de Mécanique de Lille Université des Sciences et Technologies de Lille Avenue Paul Langevin, Cité Scientifique 59655 Villeneuve d’Ascq, France.

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SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE FORMULATION AND MODAL ANALYSIS

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  1. SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE FORMULATION AND MODAL ANALYSIS Ahlem ALIA presented by Nicolas AQUELET Laboratoire de Mécanique de Lille Université des Sciences et Technologies de Lille Avenue Paul Langevin, Cité Scientifique 59655 Villeneuve d’Ascq, France

  2. Introduction The vibrations generate sound Structure Fluid The sound engenders vibrations Main industrial concern in Vibroacoustics: Reduction of NOISE Actually, noise constitutes an important indicator of quality in many industrial products such as vehicles, machinery…

  3. Introduction • Analytical technique • Simple geometry • Under very restrictive hypothesis • Numerical methods • FEM / FEM • FEM / BEM

  4. DOF f Introduction • Classical FEM / FEM • Six Nodes / Wavelength  Application domain: Low Frequency Range

  5. Introduction • FEM / FEM with Modal analysis • Modal analysis solves the vibroacoustic problem with some modes Reduction of the problem size • The modal analysis is applied with a Lumped mass representation • Lumped Mass matrix consists of Zero-off diagonal terms • Advantage of this approach: • Reduction of the computational cost (100 modes in our problem versus 432 physical unknowns)

  6. Mechanical load Simply supported elastic plate Rigid wall Introduction • Model the vibroacoustic behavior of an acoustic cavity with one flexible wall boundary by using FEM/FEM with: • Modal analysis • Lumped mass representation

  7. Structure (s) (sf) Fluid(f) (f) Vibroacoustic problem Governing equations • Structure • Fluid (1) (2) BC at Coupling Interface P: pressure k=/c wave number w: displacement : stress n: interface normal • Pressure Continuity • Normal Displacement Continuity (3) (4)

  8. Coupling system The application of the FEM to the variational formulation of structure cavity system yields to the following linear system : (5) (6) M K Ks, Ms: structural stiffness and mass matrices Kf, Mf: fluid matrices B: coupling matrix Fs, Ff: mechanical load, acoustical sources c: sound velocity, f: fluid density Ns, Nf : structural and fluid shape function

  9. (K - 2 M ) = F Coupling system • For a great number of DOF, solving the system directly is always hard in term of CPU time. • The problem (6) can be seen as an eigenvalue problem: (6)

  10. (K - 2 M ) = F Purpose of the approach We search two matrices L and R verifying: (7) (8) - L and R contain the LEFT and RIGHT eigenvectors, respectively (9) -  is a diagonal matrix containing the eigenvalues of: (10) We obtain the physical unknowns of (6) (11) by this relation:

  11. ì ü w (K - 2 M ) = F í ý p î þ Purpose of the approach Since (6) is non-symmetric, • Efficient eigenvalue algorithms can’t be used • A symmetric form of eigenvalue problem is required Sandberg’s method enables us to make it symmetric by using Modal analysis

  12. Solved Independently Classical modal analysis Cavity with stiff boundaries Structure in vacuum (12) (13) s , f : the structural and the fluid eigenvalues. Xs , Xf : the structural and the fluid eigenvectors.

  13. Classical modal analysis (50 modes) (50 modes) s, f are matrices containing some eigenvectors of structure and fluid (14) s, f verify the following properties: (15) Ds , Df are diagonal matrices containing the structural and the fluid eigenvalues. s, f represent the modal structural displacement and the modal fluid pressure.

  14. Classical modal analysis Hence, the coupling system (5) can be rewritten as the reduced system (17): (5) w , p s, f (14) 100 modal unknowns 432 physical unknowns (17) The system is reduced (the problem size is divided by 4) but it remains non symmetric

  15. Modal analysis (Sandberg Method) Non Symmetric system (17) Transition matrix (18) Symmetric system (19)

  16. Modal analysis (Sandberg Method) (19)  (20) Symmetric generalized eigenvalue problem (21) V : right eigenvector matrix of the symmetric system

  17. w , p s, f s, f Modal analysis (Sandberg Method) (18) (14) (22) (9) R: right eigenvector matrix of the original system The left eigenvector matrix “ L” is obtained in the same manner

  18. (K - 2 M ) = F Modal analysis (Sandberg Method) (6)  (11) & (9)  is a diagonal matrix containing the eigenvalues of: (10)

  19. Elastic Structure Coupling Interface Rigid Structure Fluid Numerical results

  20. Plate (b) Cavity (c) Numerical results • Discrete Kirchhoff Quadrilateral (DKQ) plate element thin plate • Kirchhoff theory • 8-node brick isoparametric acoustic element

  21. Structure ( Frequency response) • Simply supported plate (0.5m 0.5m) • Unit punctual force (0.125m 0.125m) Numerical results Results given by Migeot et al (1) Variation of the displacement with the frequency at the load point (1) 2nd Worldwide Automotive Conference Papers,1-7

  22. Natural frequencies of the plate Structure ( Natural frequencies) Structure: Simply supported plate (0.2m 0.2m) made of brass Consistent and lumped mass matrices are in good agreement with analytical ones as long as low frequencies are considered (<50th mode).

  23. Cavity ( Natural frequencies) Rigid cavity (0.2m 0.2m 0.2m) Natural frequencies of the rigid cavity FEM leads to good results below the 50th mode

  24. Field point Simply supported elastic plate Coupling problem Results Given by Lee et al (2) Numerical results Pressure at the point (0.1,0.1,0.2) CPU Time (2) Engineering Analysis with Boundary Elements, 16 (1995) 305-315

  25. Structure ( Frequency response) In vaccum Plate-cavity (air) Coupling effect 854Hz Plate quadratic displacement of the structure

  26. Mean square velocity Mean square pressure Frequency (Hz) Frequency (Hz) Coupling problem (air) Mean square pressure: cavity Mean square velocity:structure Comparison between the direct and the modal results

  27. Mean square pressure Mean square velocity Frequency (Hz) Frequency (Hz) Coupling problem (water) Mean square pressure: cavity Mean square velocity: structure Mean square velocity Comparison between the direct and the modal results

  28. Pressure (dB) Frequency (Hz) Field point Simply supported elastic plate FEM / FEM -- FEM / BEM FEM-FEM FEM-BEM comparison

  29. Conclusion • FEM / FEM with modal analysis and lumped mass representation has been used to model a simple vibroacoustic problem. • Agood representation of the mass is very essential to achieve accurate results. • Modal FEM / FEM with only small number of modes is less efficient for strong coupling. • More modes must be taken into account ( disadvantage) • Solution: Improve the numerical results by using Modal correction for diagonal system

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