Stability and numerical results for a suspension bridge of Timoshenko type with second sound

In this paper, we discuss the asymptotic behavior of a linear problem that describes the vibrations of a coupled suspension bridge. The single-span road-bed is modeled as an extensible thermoelastic damped beam, which is simply supported at the ends. The heat conduction is governed by Cattaneo’s law. The main cable is modeled as a damped string and is connected to the road-bed by a distributed system of one-sided elastic springs. First, utilizing the theory of semigroups, we prove the existence and uniqueness of the solution. Second, by constructing an appropriate Lyapunov functional, we establish exponential stability using the energy method. Numerically, we introduce fully discrete approximations based on the finite-element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Then, we show that the discrete energy decays, and a priori error estimates are established. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution.