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.

在本文中，我们讨论描述耦合悬索桥振动的线性问题的渐近行为。单跨路基被建模为可延伸的热弹性阻尼梁，其端部采用简支。热传导遵循 Cattaneo 定律。主缆被建模为阻尼弦，并通过单侧弹性弹簧分布式系统连接到路基。首先，利用半群理论证明了解的存在性和唯一性。其次，通过构造适当的李亚普诺夫函数，我们使用能量方法建立了指数稳定性。在数值上，我们引入基于有限元方法的完全离散近似来近似空间变量和隐式欧拉方案来离散时间导数。然后，我们表明离散能量衰减，并建立先验误差估计。最后，进行了一些数值模拟，以显示算法的准确性和解决方案的行为。