Torsional stability for an asymmetric system of PDEs modeling the dynamics of bridges
Several nonlinear nonlocal PDE systems for fish-bone plates have been recently proposed to model the dynamics of multi-span bridges. A fish-bone plate is composed of a central beam moving in the vertical direction, and a continuum of cross sections rotating around their center located on the beam.
The resulting system then features two degrees of freedom: the vertical displacement of the beam, governed by a beam-type equation, and the torsional angle of the cross sections, described by a wave-type equation. Building upon this framework, in this talk we introduce a PDE system of the same type but involving a nonlinearity driven by an asymmetric potential. This aims to explicitly model the slackening effect (the loss of tension) of the hangers and to analyze its impact on the stability of the structure.
After describing the model and discussing the functional setting and the weak well-posedness of the system, the seminar focus on the class of bimodal solutions, namely those where each component is concentrated on a single oscillation mode. In this regime, the torsional component satisfies an asymmetric Hill equation whose periodic coefficient is determined by the solution of an asymmetric Duffing equation. Then it'll provide some results and insights regarding the stability scenario for this equation, analyzing the differences and the complications that arise with respect to the symmetric case. The discussion will conclude by offering an overview of the stability analysis in more general situations, where multiple torsional modes are involved and the vertical component is no longer restricted to a single mode, taking also into account damping effects and external forces.
The talk is based on joint work with Maurizio Garrione (Politecnico di Milano)
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