×

zbMATH — the first resource for mathematics

Coupled string-beam equations as a model of suspension bridges. (English) Zbl 1059.74522
The authors extend results of a recent paper by G. Tajčová [Appl. Math., Praha 42, No. 6, 451–480 (1997; Zbl 1042.74535)] to a broader class of input parameters. A system of two nonlinear partial differential equations represents a model of forced vibration of a central span of a suspension bridge, i.e. of the roadbed connected with the main cable by a system of vertical stays. Using a fairly sophisticated functional analytical method, the authors prove (i) the existence of a solution for any square-integrable loading and (ii) the uniqueness of the solution and some regularity, provided the external forcing terms are small in a certain sense and the weight of the cable is sufficiently small in comparison with the weight of the roadbed.

MSC:
74H45 Vibrations in dynamical problems in solid mechanics
35Q72 Other PDE from mechanics (MSC2000)
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] J. M. Alonso, R. Ortega: Global asymptotic stability of a forced Newtonian system with dissipation. J. Math. Anal. Applications 196 (1995), 965-986. · Zbl 0844.34047
[2] J. Berkovits, P. Drábek, H. Leinfelder, V. Mustonen, G. Tajčová: Time-periodic oscillations in suspension bridges: existence of unique solutions. Nonlinear Analysis, Theory, Methods & Applications. · Zbl 0989.74031
[3] Q. H. Choi, K. Choi, T. Jung: The existence of solutions of a nonlinear suspension bridge equation. Bull. Korean Math. Soc. 33 (1996), 503-512. · Zbl 0877.35005
[4] P. Drábek: Jumping nonlinearities and mathematical models of suspension bridges. Acta Math. Inf. Univ. Ostraviensis 2 (1994), 9-18. · Zbl 0867.35006
[5] P. Drábek: Nonlinear noncoercive equations and applications. Z. Anal. Anwendungen 1 (1983), 53-65.
[6] S. Fučík, A. Kufner: Nonlinear Differential Equations. Elsevier, Holland, 1980.
[7] A. Fonda, Z. Schneider, F. Zanolin: Periodic oscillations for a nonlinear suspension bridge model. J. Comput. Appl. Math. 52 (1994), 113-140. · Zbl 0810.73030
[8] J. Glover, A. C. Lazer, P. J. McKenna: Existence and stability of large scale nonlinear oscillations in suspension bridges. J. Appl. Math. Physics (ZAMP) 40 (1989), 172-200. · Zbl 0677.73046
[9] A. Kufner, O. John, S. Fučík: Function Spaces. Academia, Prague, 1977.
[10] A. C. Lazer, P. J. McKenna: Fredholm theory for periodic solutions of some semilinear P.D.Es with homogeneous nonlinearities. Contemporary Math. 107 (1990), 109-122. · Zbl 0712.35065
[11] A. C. Lazer, P. J. McKenna: A semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities. Proc. Amer. Math. Society 106 (1989), 119-125. · Zbl 0683.34027
[12] A. C. Lazer, P. J. McKenna: Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities. Trans. Amer. Math. Society 315 (1989), 721-739. · Zbl 0725.34042
[13] A. C. Lazer, P. J. McKenna: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Review 32 (1990), 537-578. · Zbl 0725.73057
[14] A. C. Lazer, P. J. McKenna: Large scale oscillatory behaviour in loaded asymmetric systems. Ann. Inst. Henri Poincaré, Analyse non lineaire 4 (1987), 244-274. · Zbl 0633.34037
[15] P. J. McKenna, W. Walter: Nonlinear oscillations in a suspension bridge. Arch. Rational Mech. Anal. 98 (1987), 167-177. · Zbl 0676.35003
[16] M. H. Protter, H. F. Weinberger: Maximum Principles in Differential Equations. Springer-Verlag New York, 1984. · Zbl 0549.35002
[17] G. Tajčová: Mathematical models of suspension bridges. Appl. Math. 42 (1997), 451-480. · Zbl 1042.74535
[18] O. Vejvoda et al.: Partial Differential Equations-time periodic solutions. Sijthoff Nordhoff, The Netherlands, 1981.
[19] J. Weidmann: Linear Operators in Hilbert Spaces. Springer-Verlag, New York-Heidelberg-Berlin 1980. · Zbl 1025.47001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.