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Stability of a suspension bridge with a localized structural damping

  • * Corresponding author: Mohammad Al-Gharabli

    * Corresponding author: Mohammad Al-Gharabli 

The second and third authors are supported by KFUPM-Project #SB201003

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  • Strong vibrations can cause lots of damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exist evidences that several other bridges collapsed for the same reason. To overcome unwanted vibrations and prevent structures from resonating during earthquakes, winds, ..., features and modifications such as dampers are used to stabilize these bridges. In this work, we use a minimum amount of dissipation to establish exponential decay- rate estimates to the following nonlocal evolution equation

    $ u_{tt}(x,y,t)+\Delta^2 u(x,y,t) - \phi(u) u_{xx}- \left(\alpha(x, y) u_{xt}(x,y,t)\right)_x = 0, $

    which models the deformation of the deck of either a footbridge or a suspension bridge.

    Mathematics Subject Classification: Primary: 35L51, 35L71; Secondary: 35B35, 35B41.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Function $ \psi $

    Figure 2.  Smooth function $ \eta. $

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