Structural Engineering and Mechanics

  • 제12권2호
  • /
  • Pages.215-230
  • /
  • 2001
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Vibrations of long repetitive structures by a double scale asymptotic method

  • Daya, E.M. (Laboratoire de Physique et Mecanique des Materiaux UMR CNRS 7554, Institut Superieur de Genie Mecanique et Productique Universite de Metz) ;
  • Potier-Ferry, M. (Laboratoire de Physique et Mecanique des Materiaux UMR CNRS 7554, Institut Superieur de Genie Mecanique et Productique Universite de Metz)
  • 발행 : 2001.08.25
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, an asymptotic two-scale method is developed for solving vibration problem of long periodic structures. Such eigenmodes appear as a slow modulations of a periodic one. For those, the present method splits the vibration problem into two small problems at each order. The first one is a periodic problem and is posed on a few basic cells. The second is an amplitude equation to be satisfied by the envelope of the eigenmode. In this way, one can avoid the discretisation of the whole structure. Applying the Floquet method, the boundary conditions of the global problem are determined for any order of the asymptotic expansions.

키워드

참고문헌

  1. Anderson, M.S. and William, F.W. (1986), "Natural vibration and buckling of general periodic lattice structures",AIAA J., 24(1), 163-169. https://doi.org/10.2514/3.9237
  2. Bourgeois, S. (1997), "Modélisation numérique des panneaux structuraux legers", Thesis of University of Aix-MarseilleII-France.
  3. Brillouin, L. (1953), Wave Propagation in Periodic Structures, Dover, New York.
  4. Caillerie, D. (1989), "Thin elastic and periodic plates", Math. Meth. in Appl. Sci., 6, 159-191.
  5. Caillerie, D., Trompette, P. and Verna, P. (1989), "Homogeneization of periodic trusses", Congress IASS, Madrid.
  6. Castanier, M.P. and Pierre, C. (1995), "Lyapunov exponents and localization phenomena in multi-coupled nearlyperiodic systems", J. Sound and Vibration 183(3), 493-515. https://doi.org/10.1006/jsvi.1995.0267
  7. Damil, N. and Potier-Ferry, M. (1990), "A new method to compute perturbed bifurcation: Application to thebuckling of imperfect elastic structures", Int. J. of Eng. Sci., 28(9), 943-957. https://doi.org/10.1016/0020-7225(90)90043-I
  8. Daya, E.M. (1994), "Vibration et stabilite des longues structures flexibles a forme repetitive", Thesis of Universityof Metz-France.
  9. Faulkner, M.G. and Hongo, D.P. (1985), "Free vibrations of a mono-coupled periodic system", J. Sound andVibration, 99(1), 29-42. https://doi.org/10.1016/0022-460X(85)90443-2
  10. Flotow, A.H. von (1986), "Disturbance propagation in structural networks", J. Sound and Vibration, 106(3), 99-118.
  11. John, O.W., Su, Z.W. and Feng, C.C. (1985), "Equivalent continuum representation of structures composed ofrepeated elements", AIAA J. 23(10), 1564-1569. https://doi.org/10.2514/3.9124
  12. Jordan, D.W. and Smith, P. (1987), "Non-linear ordinary differential equations", 2nd edn., Clarendon Press, Oxford.
  13. Lee, S.Y. and Ke, H.Y. (1992), "Flexural wave propagation in an elastic beam with periodic structure", J. Appl.Mech., 57, 779-783.
  14. Lin, Y.L. (1962), "Free vibration of continous beam on elastic supports", Int. J. Mech. Sci., 4, 409-423. https://doi.org/10.1016/S0020-7403(62)80027-7
  15. Lin, Y.L. and McDaniel, T.J. (1969), "Dynamics of Beam-type periodic structures", J. Eng. Industry, Nov, 1133-1141.
  16. McDaniel, T.J. and Chang, K.J. (1980), "Dynamics of rotationnally periodic large space structures", J. Soundand Vibration, 66, 351-368.
  17. Mead, D.J. (1970), "Free wave propagation in periodically support infinite beam", J. Sound and Vibration, 11(2),81-197.
  18. Moreau, G. (1996), Thesis of university of Grenoble-France. Homogeneisation de structures discretes en elasticite eten incremental. Application aux modelisations continues lineaires et non-lineaires de treillis quasi-periodiques.
  19. Noor, A.K. (1988), "Continuum modeling for repetitive lattice structures", Appl. Mech. Rev., 41(7), 285-296. https://doi.org/10.1115/1.3151907
  20. Noor, A.K. and Andersen, C.M. (1979), "Analysis of beamlike lattice trusses", Comput. Meth. Appl. Mech. andEng., 20, 53-70. https://doi.org/10.1016/0045-7825(79)90058-6
  21. Noor, A.K. and Anderson, M.S. (1970), "Continuum models for beam and plate-like lattice structures", AmericanInst. of Aeronaut. and Astronaut. J. 16, 1219-1228.
  22. Noor, A.K. and Nemeth, M.P. (1980), "Analysis of spatial beamlike lattices with rigid joints", Computer. Meth.Appl. Mech. and Eng., 24, 35-59. https://doi.org/10.1016/0045-7825(80)90039-0
  23. Sanchez Hubert, J. and Sanchez Palencia, E. (1989), Vibrating and Coupling Continuous Systems, Asymptotic. methods, Springer-Verlag, Berlin.
  24. Sen Gupta, G. (1970), "Natural flexural waves and the normal modes of periodically-supported beams andplates", J. Sound and Vibration 13(1), 89-101. https://doi.org/10.1016/S0022-460X(70)80082-7
  25. Touratier, M. (1986), "Floquet waves in a body with slender periodic structure", Wave Motion, 8, 485-495. https://doi.org/10.1016/0165-2125(86)90032-6
  26. Von Flotow, A.H. (1986), "Disturbance propagation in structural networks", J. Sound and Vibration, 106(3), 433-450. https://doi.org/10.1016/0022-460X(86)90190-2
  27. Wesfreid, J.E. and Zaleski, S. (1984), "Cellular structures in instability problems", Lecture Notes in Physics,Spring, Berlin.
  28. William, F.W. (1986), "Exact eigenvalue calculations for structures with rotationally periodic substructures", Int.J. Numer. Meth. Eng., 24(4), 695-706.
  29. Young, Y. and Lin, Y.L. (1989), "Propagation of decaying waves in periodic and piecewise periodic structures offinite length", J. Sound and Vibration, 129(2), 99-118. https://doi.org/10.1016/0022-460X(89)90538-5

피인용 문헌

  1. Modelisation of modulated vibration modes of repetitive structures vol.168, pp.1-2, 2004, https://doi.org/10.1016/j.cam.2003.07.001
  2. Continuum modeling for the modulated vibration modes of large repetitive structures vol.330, pp.5, 2002, https://doi.org/10.1016/S1631-0721(02)01464-X
  3. Evaluation of continuous modelings for the modulated vibration modes of long repetitive structures vol.44, pp.21, 2007, https://doi.org/10.1016/j.ijsolstr.2007.03.023
  4. A two scale method for modulated vibration modes of large, nearly repetitive, structures vol.331, pp.6, 2003, https://doi.org/10.1016/S1631-0721(03)00093-7
  5. Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure vol.386, 2017, https://doi.org/10.1016/j.jsv.2016.09.025
  6. Elastic waves in periodically heterogeneous two-dimensional media: locally periodic and anti-periodic modes vol.474, pp.2215, 2018, https://doi.org/10.1098/rspa.2017.0908
  7. Composite dynamic models for periodically heterogeneous media pp.1741-3028, 2018, https://doi.org/10.1177/1081286518776704