Identification of Reduced Models from Optimal Complex Eigenvectors in Structural Dynamics and Vibroacoustics

Abstract

The objective of this chapter is to present some efficient techniques for identification of reduced models from experimental modal analysis in the fields of structural dynamics and vibroacoustics. The main objective is to build mass, stiffness and damping matrices of an equivalent system which exhibits the same behavior as the one which has been experimentally measured. This inverse procedure is very sensitive to experimental noise and instead of using purely mathematical regularization techniques, physical considerations can be used. Imposing the so-called properness condition of complex modes on identified vectors leads to matrices which have physical meanings and whose behavior is as close as possible to the measured one. Some illustrations are presented on structural dynamics. Then the methodology is extended to vibroacoustics and illustrated on measured data.

Keywords

Structural Dynamic Complex Mode Matrice Identification Left Eigenvector Inverse Procedure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The authors would like to thank Jean-Loïc Le Carrou from Laboratoire d’Acoustique Musicale (Paris VI) and François Gautier from the Laboratoire d’Acoustique de l’Université du Maine, for the fruitful discussions and for allowing us to use their measurements data, used in the last part of the chapter.

References

  1. 1.
    Adhikari, S.: Optimal complex modes and an index of damping non-proportionality. Mech. Syst. Signal Process. 18(1), 1–28 (2004) CrossRefGoogle Scholar
  2. 2.
    Adhikari, S.: Damping modelling using generalized proportional damping. J. Sound Vib. 293(1–2), 156–170 (2006) CrossRefGoogle Scholar
  3. 3.
    Adhikari, S., Woodhouse, J.: Identification of damping: Part 1, viscous damping. J. Sound Vib. 243(1), 43–61 (2001) CrossRefGoogle Scholar
  4. 4.
    Adhikari, S., Woodhouse, J.: Identification of damping: Part 2, non-viscous damping. J. Sound Vib. 243(1), 63–88 (2001) CrossRefGoogle Scholar
  5. 5.
    Adhikari, S., Woodhouse, J.: Identification of damping: Part 3, symmetry-preserving methods. J. Sound Vib. 251(3), 477–490 (2002) CrossRefGoogle Scholar
  6. 6.
    Adhikari, S., Woodhouse, J.: Identification of damping: Part 4, error analysis. J. Sound Vib. 251(3), 491–504 (2002) CrossRefGoogle Scholar
  7. 7.
    Balmès, E.: New results on the identification of normal modes from experimental complex ones. Mech. Syst. Signal Process. 11(2), 229–243 (1997) CrossRefGoogle Scholar
  8. 8.
    Barbieri, N., Souza Júnior, O.H., Barbieri, R.: Dynamical analysis of transmission line cables. Part 2—damping estimation. Mech. Syst. Signal Process. 18(3), 671–681 (2004) CrossRefGoogle Scholar
  9. 9.
    Bernal, D., Gunes, B.: Extraction of second order system matrices from state space realizations. In: 14th ASCE Engineering Mechanics Conference (EM2000), Austin, Texas (2000) Google Scholar
  10. 10.
    Bert, C.W.: Material damping: An introductory review of mathematic measures and experimental technique. J. Sound Vib. 29(2), 129–153 (1973) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Caughey, T.K., O’Kelly, M.E.: Classical normal modes in damped linear systems. J. Appl. Mech. 32, 583–588 (1965) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, S.Y., Ju, M.S., Tsuei, Y.G.: Estimation of mass, stiffness and damping matrices from frequency response functions. J. Vib. Acoust. 118(1), 78–82 (1996) CrossRefGoogle Scholar
  13. 13.
    Christensen, O., Vistisen, B.B.: Simple model for low frequency guitar function. J. Acoust. Soc. Am. 68(3), 758–766 (1980) CrossRefGoogle Scholar
  14. 14.
    Craig, R.J., Bampton, M.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968) MATHCrossRefGoogle Scholar
  15. 15.
    Crandall, S.H.: The role of damping in vibration theory. J. Sound Vib. 11(1), 3–18 (1970) MATHCrossRefGoogle Scholar
  16. 16.
    Everstine, G.C. Finite element formulations of structural acoustics problems. Comput. Struct. 65(3), 307–321 (1997) MATHCrossRefGoogle Scholar
  17. 17.
    Fillod, R., Piranda, J.: Research method of the eigenmodes and generalized elements of a linear mechanical structure. Shock Vib. Bull. 48(3), 5–12 (1978) Google Scholar
  18. 18.
    Fletcher, N.H., Rossing, T.D.: The Physics of Musical Instruments. Springer, Berlin (1998) MATHGoogle Scholar
  19. 19.
    Fritzen, C.P.: Identification of mass, damping, and stiffness matrices of mechanical systems. J. Vib. Acoust. Stress Reliab. Des. 108, 9–16 (1986) CrossRefGoogle Scholar
  20. 20.
    Gaul, L.: The influence of damping on waves and vibrations. Mech. Syst. Signal Process. 13(1), 1–30 (1999) CrossRefGoogle Scholar
  21. 21.
    Ibrahim, S.R.: Dynamic modeling of structures from measured complex modes. AIAA J. 21(6), 898–901 (1983) CrossRefGoogle Scholar
  22. 22.
    Ibrahim, S.R., Sestieri, A.: Existence and normalization of complex modes in post experimental use in modal analysis. In 13th International Modal Analysis Conference, Nashville, USA, pp. 483–489 (1995) Google Scholar
  23. 23.
    Kasai, T., Link, M.: Identification of non-proportional modal damping matrix and real normal modes. Mech. Syst. Signal Process. 16(6), 921–934 (2002) CrossRefGoogle Scholar
  24. 24.
    Lancaster, P., Prells, U.: Inverse problems for damped vibrating systems. J. Sound Vib. 283(3–5), 891–914 (2005) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Le Carrou, J.-L., Gautier, F., Foltête, E.: Experimental study of A0 and T1 modes of the concert harp. J. Acoust. Soc. Am. 121(1), 559–567 (2007) CrossRefGoogle Scholar
  26. 26.
    Lee, J.H., Kim, J.: Development and validation of a new experimental method to identify damping matrices of a dynamic system. J. Sound Vib. 246(3), 505–524 (2001) CrossRefGoogle Scholar
  27. 27.
    Lin, R.M., Zhu, J.: On the relationship between viscous and hysteretic damping models and the importance of correct interpretation for system identification. J. Sound Vib. 325(1–2), 14–33 (2009) CrossRefGoogle Scholar
  28. 28.
    Minas, C., Inman, D.J.: Identification of a nonproportional damping matrix from incomplete modal information. J. Vib. Acoust. 113(2), 219–224 (1991) CrossRefGoogle Scholar
  29. 29.
    Morand, H.J.-P., Ohayon, R.: Fluid Structure Interaction. Wiley, New York (1995) MATHGoogle Scholar
  30. 30.
    Ouisse, M., Foltête, E.: On the comparison of symmetric and unsymmetric formulations for experimental vibro-acoustic modal analysis. In: Acoustics’08, Paris, 2008 Google Scholar
  31. 31.
    Ozgen, G.O., Kim, J.H.: Direct identification and expansion of damping matrix for experimental-analytical hybrid modeling. J. Sound Vib. 308(1–2), 348–372 (2007) CrossRefGoogle Scholar
  32. 32.
    Pilkey, D.F., Park, G., Inman, D.J.: Damping matrix identification and experimental verification. In: Smart Structures and Materials, SPIE Conference on Passive Damping and Isolation, Newport Beach, California, pp. 350–357 (1999) Google Scholar
  33. 33.
    Prandina, M., Mottershead, J.E., Bonisoli, E.: An assessment of damping identification methods. J. Sound Vib. 323(3–5), 662–676 (2009) CrossRefGoogle Scholar
  34. 34.
    Rayleigh, J.W.S. The Theory of Sound, vols. 1, 2. Dover, New York (1945) Google Scholar
  35. 35.
    Srikantha Phani A., Woodhouse, J.: Viscous damping identification in linear vibration. J. Sound Vib. 303(3–5), 475–500 (2007) CrossRefGoogle Scholar
  36. 36.
    Srikantha Phani A., Woodhouse, J.: Experimental identification of viscous damping in linear vibration. J. Sound Vib. 319(3–5), 832–849 (2009) CrossRefGoogle Scholar
  37. 37.
    Tran, Q.H., Ouisse, M., Bouhaddi, N.: A robust component mode synthesis method for stochastic damped vibroacoustics. Mech. Syst. Signal Process. 24(1), 164–181 (1997) CrossRefGoogle Scholar
  38. 38.
    Van der Auweraer, H., Guillaume, P., Verboven, P., Vanlanduit, S.: Application of a fast-stabilizing frequency domain parameter estimation method. J. Dyn. Syst. Meas. Control 123(4), 651–658 (2001) CrossRefGoogle Scholar
  39. 39.
    Wyckaert, K., Augusztinovicz, F., Sas, P.: Vibro-acoustical modal analysis: reciprocity, model symmetry and model validity. J. Acoust. Soc. Am. 100(5), 3172–3181 (1996) CrossRefGoogle Scholar
  40. 40.
    Xu, J.: A synthesis formulation of explicit damping matrix for non-classically damped systems. Nucl. Eng. Des. 227(2), 125–132 (2004) CrossRefGoogle Scholar
  41. 41.
    Zhang, Q., Lallement, G.: Comparison of normal eigenmodes calculation methods based on identified complex eigenmodes. J. Spacecr. Rockets 24, 69–73 (1987) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.FEMTO-ST Institute, Applied MechanicsUniversity of Franche-ComtéBesançonFrance

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