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In this work we try to explain various mathematical models describing the dynamical behaviour of suspension bridges such as the Tacoma Narrows bridge. Our attention is concentrated on the derivation of these models, an interpretation of particular parameters and on a discussion of their advantages and disadvantages. Our work should be a starting point for a qualitative study of dynamical structures of this type and that is why we have a closer look at the models, which have not been studied in literature...
In this paper we present several nonlinear models of suspension bridges; most of them have been introduced by Lazer and McKenna. We discuss some results which were obtained by the authors and other mathematicians for the boundary value problems and initial boundary value problems. Our intention is to point out the character of these results and to show which mathematical methods were used to prove them instead of giving precise proofs and statements.
Two mathematical models of railway track oscillations are compared on the basis of numerical experiments.
A nonlinear model of the vibrating string is studied and global existence and uniqueness theorems for the solution of the Cauchy-Dirichlet problem are given. The model is then compared to the classical D'Alembert model and to a nonlinear model due to Kirchhoff.
The vibration problem in two variables is derived from the spatial situation (a plate as a three-dimensional body) on the basis of geometrically nonlinear plate theory (using Kármán's hypothesis) and coupled linear thermoelasticity. That leads to coupled strongly nonlinear two-dimensional equilibrium and heat conducting equations (under classical mechanical and thermal boundary conditions).
For the generalized problem with subgradient conditions on the boundary and in the domain (including also...
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