A mathematical model of suspension bridges. (English) Zbl 1099.74037
The author considers a one-dimensional nonlinear string-beam system describing the vertical oscillations of a suspension bridge which is coupled with the main cable by the stays. The main cable is modelled as a vibrating string and the roadbed of the bridge is represented by a bending beam with simply supported ends. Using the Galerkin method together with the Leray-Schauder principle, the existence of a weak periodic solution is proved. Moreover, a regularity result is also given under some additional assumptions.
Reviewer: Petr Nečesal (Plzeň)
MSC:
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 35B10 | Periodic solutions to PDEs |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 35Q72 | Other PDE from mechanics (MSC2000) |
| 35A35 | Theoretical approximation in context of PDEs |
References:
| [1] | J. M. Ball: Initial-boundary value problems for an extensible beam. J. Math. Anal. Appl. 42 (1973), 61-90. · Zbl 0254.73042 · doi:10.1016/0022-247X(73)90121-2 |
| [2] | Q. H. Choi, T. Jung: On periodic solutions of the nonlinear suspension bridge equation. Differ. Integral Equ. 4 (1991), 383-396. · Zbl 0733.35104 |
| [3] | P. Drábek: Jumping nonlinearities and mathematical models of suspension bridges. Acta Math. et Inf. Univ. Ostraviensis 2 (1994), 9-18. · Zbl 0867.35006 |
| [4] | P. Drábek, H. Leinfelder, and G. Tajčová: Coupled string-beam equations as a model of suspension bridges. Appl. Math. 44 (1999), 97-142. · Zbl 1059.74522 · doi:10.1023/A:1022257304738 |
| [5] | N. Krylová: Periodic solutions of hyperbolic partial differential equation with quadratic dissipative term. Czechoslovak Math. J. 20(95) (1970), 375-405. · Zbl 0214.10402 |
| [6] | 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. · Zbl 0725.73057 · doi:10.1137/1032120 |
| [7] | P. J. McKenna, W. Walter: Nonlinear oscillations in a suspension bridge. Arch. Ration. Mech. Anal. 98 (1987), 167-177. · Zbl 0676.35003 · doi:10.1007/BF00251232 |
| [8] | P. J. McKenna, W. Walter: Travelling waves in a suspension bridge. SIAM J. Appl. Math. 50 (1990), 703-715. · Zbl 0699.73038 · doi:10.1137/0150041 |
| [9] | L. Sanchez: Periodic solutions of a nonlinear evolution equation with a linear dissipative term. Rend. Sem. Mat. Univ. Politec. Torino 37 (1980), 183-191. · Zbl 0459.35009 |
| [10] | G. Tajčová: Mathematical models of suspension bridges. Appl. Math. 42 (1997), 451-480. · Zbl 1042.74535 · doi:10.1023/A:1022255113612 |
| [11] | E. Zeidler: Nonlinear Functional Analysis and Its Applications, Vol. I-III. Springer-Verlag, New York, 1985-1990. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
