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cable stayed bridge; vertical and torsional oscillations; eigenvalues and eigenfunctions of center span
In this paper the stability of two basic types of cable stayed bridges, suspended by one or two rows of cables, is studied. Two linearized models of the center span describing the vertical and torsional oscillations are investigated. After the analysis of these models, a stability criterion is formulated. The criterion expresses a relation between the eigenvalues of the vertical and torsional oscillations of the center span. The continuous dependence of the eigenvalues on some data is studied and a stability problem for the center span is formulated. The existence of a solution to the stability problem is proved. Some other qualitative results concerning the stability/instability of oscillations are studied as well.
[1] N. U.  Ahmed, H.  Harbi: Mathematical analysis of dynamic models of suspension bridges. SIAM J.  Appl. Math. 58 (1998), 853–874. DOI 10.1137/S0036139996308698 | MR 1616611
[2] J.  Berkovits, P.  Drábek, H.  Leinfelder, V.  Mustonen, and G.  Tajčová: Time-periodic oscillations in suspension bridges: Existence of unique solution. Nonlinear Anal., Real World Appl. 1 (2000), 345–362. MR 1791531
[3] P.  Drábek, H.  Leinfelder, G. Tajčová: Coupled string-beam equations as a model of suspension bridges. Appl. Math. 44 (1999), 97–142. DOI 10.1023/A:1022257304738 | MR 1667633
[4] A.  Fonda, Z.  Schneider, and F.  Zanolin: Periodic oscillations for a nonlinear suspension bridge model. J.  Comput. Appl. Math. 52 (1994), 113–140. DOI 10.1016/0377-0427(94)90352-2 | MR 1310126
[5] J.  Glover, A. C.  Lazer, and P. J.  McKenna: Existence and stability of large-scale nonlinear oscillations in suspension bridges. Z. Angew. Math. Phys. 40 (1989), 171–200. DOI 10.1007/BF00944997 | MR 0990626
[6] A.  Kufner, O.  John, S. Fučík: Functional spaces. Academia, Prague, 1977.
[7] A. C.  Lazer, P. J.  McKenna: Large scale oscillatory behaviour in loaded asymmetric systems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4 (1987), 243–274. DOI 10.1016/S0294-1449(16)30368-7 | MR 0898049
[8] A. C.  Lazer, P. J.  McKenna: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 32 (1990), 537–578. DOI 10.1137/1032120 | MR 1084570
[9] A. C.  Lazer, P. J.  McKenna: Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities. Trans. Am. Math. Soc. 315 (1989), 721–739. DOI 10.1090/S0002-9947-1989-0979963-1 | MR 0979963
[10] P. J.  McKenna, W.  Walter: Nonlinear oscillations in a suspension bridge. Arch. Rational Mech. Anal. 98 (1987), 167–190. DOI 10.1007/BF00251232 | MR 0866720
[11] P. J.  McKenna, W.  Walter: Travelling waves in a suspension bridge. SIAM J.  Appl. Math. 50 (1990), 703–715. DOI 10.1137/0150041 | MR 1050908
[12] J.  Malík: Oscillations in cable stayed bridges: existence, uniqueness, homogenization of cable systems. J.  Math. Anal. Appl. 226 (2002), 100–126. DOI 10.1006/jmaa.2001.7713 | MR 1876772
[13] J.  Malík: Mathematical modelling of cable stayed bridges: existence, uniqueness, homogenization of cable systems. Appl. Math. 49 (2004), 1–38. DOI 10.1023/B:APOM.0000024518.38660.a3 | MR 2032146
[14] E.  Simiu, R. H.  Scanlan: Wind Effects on Structures: An Introduction to Wind Engineering. John Wiley, New York, 1978.
[15] S. L.  Sobolev: Applications of Functional Analysis in Mathematical Physics. American Mathematical Society, Providence, 1963. MR 0165337 | Zbl 0123.09003
[16] G. Tajčová: Mathematical models of suspension bridges. Appl. Math. 42 (1997), 451–480. DOI 10.1023/A:1022255113612 | MR 1475052
[17] V. S.  Vladimirov: Equations of Mathematical Physics. Mir, Moscow, 1987. (Italian) MR 1018346 | Zbl 0699.35005
[18] R.  Walther, B.  Houriet, W.  Isler, P.  Moïa, and J. F. Klein: Cable Stayed Bridges. Thomas Telford, London, 1999.
[19] K.  Yosida: Functional Analysis. Springer-Verlag, Berlin-Götingen-Heidelberg, 1965. Zbl 0126.11504
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