which might also be written as Ks 0 Ksf T Kf −ω2 Ms MT fs 0 Mf φL s φL f =0 (48) The first line of this equation yields φL s = ω2 K−1 s Ms φ L s +K−1 s MT fs φ L f = ω2 K−1 s Ms φ L s −K−1 s Ksf φ L f (49) where we have used the relationMT fs =−Ksf explained in section 2.1. Substituting this relation in (46) one finds φR = 1 ω2 φL s φL f or φL = ω2 φR s φRf (50) which is inline with the proofs in [21] and [22]. References 1. Dhatt Gouri, Touzot Gilbert, The Finite Element Method Displayed, 1984, John Wiley and Sons, 1984 2. Wu T.W., Boundary Element Acoustics, Fundamentals and Computer Codes, WIT Press, 2000 3. Betess P., Infinite Elements, International Journal for Numerical Methods in Engineering, Vol.11, 53-64, 1977 4. Morand, H.J.P., Ohayon R., Fluid Structure Interaction, John Wiley and Sons, Masson, 1995 5. Craig R.R., Bampton M.C.C., Coupling of Substructures for Dynamic Analysis, AIAA Journal, vol.6(7), 1313-1319, 1968 6. Rubin S., Improved Component-Mode Representation for Structural Dynamic Analysis,AIAA Journal, vol.13(8), 995-1006, 1975 7. Felippa Carlos A., Symmetrization of the contained compressible-fluid vibration eigenproblem, Communications in Applied Numerical Methods, vol.1, 241-247, 1985 8. Felippa Carlos A., Symmetrization of the coupled eigenproblems by eigenvector augmentation, Communications in Applied Numerical Methods, vol.4(4), 561-563, 1988 9. Tabak Umut, Rixen Daniel J., Reduced iterative correction algorithm for coupled vibroacoustic problems, Proceedings of ISMA 2010, International Conference on Noise and Vibration Engineering, 4685-4695, 2010 10. Maess Mathiass, Gaul Lothar, Component Mode Synthesis for Efficient Structure-Acoustic Simulation of Piping Systems, International Modal Analysis Conference XXIV, St.Louis, 2006. 11. Nactergaele Ph., Rixen D.J.,Steenhoek A.M., Efficient weakly coupled projection basis for the reduction of thermo-mechanical models, Journal of Computational and Applied Mathematics, Volume 234, Issue 7, 2010, Pages 2272-2278 12. Ge´radin M., Rixen D., Mechanical Vibrations. Theory and Application to Structural Dynamics,Wiley & Sons,2nd Ed, 1997 13. Craig R.R., Kurdila A., Fundamentals of Structural Dynamics, John. Wiley and Sons, 2006 14. Brechlin E., Gaul L., Two methodological improvements for component mode synthesis, International Conference on Noise and Vibration Engineering, ISMA2000, 2000 Globally enriched substructuring techniques for vibro-acoustic simulation 279
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