On the well-posedness and stability for a coupled nonlinear suspension bridge problem

Suspension bridges are critical, lifeline, civil structures that have been constructed in many countries due to their superior effectiveness when it comes to long spans, in which other bridge system cannot handle, see fig.1. In this paper, we study a mathematical model for a one-dimensional suspension bridge problem with nonlinear damping. The model takes into consideration the vibration of the bridge deck in the vertical plane and main cable from which the bridge deck is suspended. The well-posedness and asymptotic stability are investigated using the well-known Galerkin and multiplier methods. Next, a numerical solution using the finite difference method, along with its stability criterion is developed to solve the system of partial differential equations governing the mathematical model of the bridge system. Numerical applications are then presented to showcase the effect of the nonlinear damping parameters and the dissipated energy time histories. Results of this numerical solution proved that an optimum damping and energy dissipation can be achieved.