Skip to main content
SpringerLink
Log in

Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Localization phenomena in one-dimensional imperfect continuous structures are analyzed, both in dynamics and buckling. By using simple models, fundamental concepts about localization are introduced and similarities between dynamics and buckling localization are highlighted. In particular, it is shown that strong localization of the normal modes is due to turning points in which purely imaginary characteristic exponents assume a non zero real part; in contrast, if turning points do not occur, only weak localization can exist. The possibility of a disturbance propagating along the structure is also discussed. A perturbation method is then illustrated, which generalizes the classical WKB method; this allows the differential problem to be transformed into a sequence of algebraic problems in which the spatial variable appears as a parameter. Applications of the method are worked out for beams and strings on elastic soil. All these structures are found to have nearly-defective system matrices, so their characteristic exponents are highly sensitive to imperfections.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Matsuda, H. and Ishii, K., 'Localization of normal modes and energy transport in the disordered harmonic chain', Supplement of the Progress of Theoretical Physics 45, 1970, 56–86.

    Google Scholar 

  2. Ishii, K., 'Localization of eigenstates and transport phenomena in the one-dimensional disordered system', Supplement of the Progress of Theoretical Physics 53, 1973, 77–138.

    Google Scholar 

  3. Hodges, C. H., 'Confinement of vibration by structural irregularity', Journal of Sound and Vibration 82, 1982, 411–424.

    Google Scholar 

  4. Li, D. and Benaroya, H., 'Dynamics of periodic and near-periodic structures', Applied Mechanics Review 45(11), 1982, 447–459.

    Google Scholar 

  5. Benaroya, H., 'Localization and effects of irregularities in structures', Applied Mechanics Review 49, 1996, 57–135.

    Google Scholar 

  6. Pierre, C. and Dowell, E. H., 'Localization of vibrations by structural irregularity', Journal of Sound and Vibration 114(3), 1987, 549–564.

    Google Scholar 

  7. Bendiksen, O., 'Mode localization phenomena in large space structures', AIAA Journal 25(9), 1987, 1241–1248.

    Google Scholar 

  8. Pierre, C., 'Mode localization and eigenvalue loci veering phenomena in disordered structures', Journal of Sound and Vibration 126(3), 1988, 485–502.

    Google Scholar 

  9. Pierre, C., 'Weak and strong vibration localization in disordered structures: A statistical investigation', Journal of Sound and Vibration 139(1), 1990, 111–132.

    Google Scholar 

  10. Wu, G., 'Free vibration of cyclic assemblies with a single disordered component', Journal of Sound and Vibration 165(3), 1993, 567–570.

    Google Scholar 

  11. Ottarsson, G. and Pierre, C., 'A transfer matrix approach to free vibration localization in mistuned blade assemblies', Journal of Sound and Vibration 197(5), 1996, 589–618.

    Google Scholar 

  12. Hodges, C. H. and Woodhouse, J., 'Vibration isolation from irregularity in a nearly periodic structure: Theory and measurements', Journal of Acoustical Society of America 79(9), 1983, 894–905.

    Google Scholar 

  13. Cornwell, P. J. and Bendiksen, O., 'Localization of vibrations in large space reflectors', AIAA Journal 27(2), 1989, 219–226.

    Google Scholar 

  14. Pierre, C. and Cha, P. D., 'Strong mode localization in nearly periodic disordered structures', AIAA Journal 27(2), 1989, 227–241.

    Google Scholar 

  15. Al-Jawi, A. A., Ulsoy, A. G., and Pierre, C., 'Vibration localization in dual-span, axially moving beams', Journal of Sound and Vibration 179(2), 1995, 243–266 (part I), 267-287(part II).

    Google Scholar 

  16. Keane, A. J. and Price, W. G., 'On the vibrations of mono-coupled periodic and near-periodic structures', Journal of Sound and Vibration 128(3), 1989, 423–450.

    Google Scholar 

  17. Cai, G. Q. and Lin, Y. K., 'Localization of wave propagation in disordered periodic structures', AIAA Journal 29, 1990, 450–456.

    Google Scholar 

  18. Lin, Y. K. and Cai, G. Q., 'Disordered periodic structures', in Computational Stochastic Mechanics, C. A. Brebbia and P. Spanos (eds.), Elsevier, Amsterdam, 1991, pp. 777–788.

    Google Scholar 

  19. Cha, P. D. and Pierre, C., 'Vibration localization by disorder in assemblies of monocoupled multimode component systems', Journal of Applied Mechanics 58, 1991, 1072–1081.

    Google Scholar 

  20. Cheng, S. P. and Perkins, N. C., 'Closed-form vibration analysis of sagged cables/mass suspension', ASME Paper 92WA/APM-7, 1992.

  21. Bouzit, D. and Pierre, C., 'Localization of vibration in disordered multi-span beams with damping', Journal of Sound and Vibration 187(4), 1995, 625–648.

    Google Scholar 

  22. Scott, J. M. F., 'The statistics of waves propagating in a one-dimensional random medium', Proceedings of Royal Society of London A398, 1985, 341–363.

    Google Scholar 

  23. Luongo, A., 'Mode localization by structural imperfections in one-dimensional continuous systems', Journal of Sound and Vibration 155(2), 1992, 249–271.

    Google Scholar 

  24. Tjavaras, A. A. and Triantafyllou, M. S., 'Non-linear response of two disordered pendula', Journal of Sound and Vibration 190(1), 1996, 65–76.

    Google Scholar 

  25. Zevin, A. A., 'Localization of periodic oscillations in discrete non-linear systems', Journal of Sound and Vibration 193(4), 1996, 847–862.

    Google Scholar 

  26. Vakakis, A., Nayfeh, T., and King, M., 'A multiple-scales analysis of nonlinear, localized modes in a cyclic periodic system', Transactions of the ASME 60, 1993, 388–397.

    Google Scholar 

  27. Emaci, E., Nayfeh, T. A. and Vakakis, A. F., 'Numerical and experimental study of nonlinear localization in a flexible structure with vibro-impacts', Journal of Applied Mathematics and Mechanics (ZAMM) 77(7), 1997, 527–541.

    Google Scholar 

  28. Nayfeh, T. A., Emaci, E. and Vakakis, A. F., 'Application of nonlinear localization to the optimization of a vibration isolation system', AIAA Journal 35(8), 1997, 1378–1386.

    Google Scholar 

  29. Pierre, C., 'Curve veering and mode localization in a buckling problem', Journal of Applied Mathematics and Physics (ZAMP) 40, 1989, 758–761.

    Google Scholar 

  30. Gioncu, V., 'General theory of coupled instabilities', Thin Walled Structures 19(2-4), 1994, 81–127.

    Google Scholar 

  31. Luongo, A., 'On the amplitude modulation and localization phenomena in interactive buckling problems', International Journal of Solids and Structures 27(15), 1991, 1943–1954.

    Google Scholar 

  32. Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973.

    Google Scholar 

  33. Tvegaard, V. and Needleman, A., 'On the localization of buckling patterns', Journal of Applied Mechanics 47, 1980, 613–619.

    Google Scholar 

  34. Tvegaard, V. and Needleman, A., 'On the development of localized buckling patterns', in Collapse, The Buckling of Structures in Theory and Practice, J. M. T. Thompson and G. W. Hunt (eds.), Cambridge University Press, Cambridge, 1983, pp. 1–17.

    Google Scholar 

  35. Potier-Ferry, M., 'Amplitude modulation, phase modulation and localization of buckling patterns', in Collapse, The Buckling of Structures in Theory and Practice, J. M. T. Thompson and G. W. Hunt (eds.), Cambridge University Press, Cambridge, 1983, pp. 149–159.

    Google Scholar 

  36. Luongo, A. and Pignataro, M., 'Multiple interaction and localization phenomena in the postbuckling of compressed thin-walled members', AIAA Journal 26(11), 1988, 1395–1402.

    Google Scholar 

  37. Hunt, G. W., Bolt, H. M., and Thompson, J. M. T., 'Structural localization phenomena and the dynamical phase-space analogy', Proceedings of Royal Society London A425, 1989, 245–267.

    Google Scholar 

  38. Luongo, A., 'Perturbation methods for the eigenvalue analysis of imperfect systems with high modal density', Parts I and II, Accademia Peloritana dei Pericolanti, Messina (Italy), Vol. LXVIII, 401–423, 1991, 424-447 [in Italian].

    Google Scholar 

  39. Wilkinson, J. H., The Algebraic Eigenvalue Problem, Oxford University Press, London, 1965.

    Google Scholar 

  40. Kato, T., A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1982.

    Google Scholar 

  41. Luongo, A., 'Eigensolutions sensitivity for nonsymmetric matrices with repeated eigenvalues', AIAA Journal 31(7), 1994, 1321–1328.

    Google Scholar 

  42. Luongo, A., 'Eigensolutions of perturbed nearly-defective matrices', Journal of Sound and Vibration 185(3), 1995, 377–395.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Structural Engineering, University of L'Aquila, 67040, Monteluco Roio, Italy

    Angelo Luongo

Authors
  1. Angelo Luongo
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luongo, A. Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures. Nonlinear Dynamics 25, 133–156 (2001). https://doi.org/10.1023/A:1012954700751

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1012954700751

Search

Navigation