Effects of Viscoelastic Damping and Nonlinear Feedback Modulated by Time-Dependent Coefficient in a Suspension Bridge System: Existence and Stability Decay Results

This work presents a mathematical model for a one-dimensional suspension bridge system, incorporating viscoelastic damping and nonlinear frictional damping with time-dependent coefficients. The model accounts for the vibrations of both the bridge deck in the vertical plane and the main cable, which supports the deck via suspenders. In this study, we investigate the interaction between a viscoelastic damping mechanism, which depends on the material memory effects, and nonlinear frictional damping, which arises from contact and surface forces, within the proposed model. First, we apply the Faedo–Galerkin method to establish the existence and uniqueness of solutions for the system. Next, using the multiplier method, we derive an explicit formula for the energy decay rate, showing that the decay behavior depends on the relaxation function, the nonlinear frictional damping and the time-dependent coefficient. Our findings are achieved without imposing restrictive growth conditions on the frictional damping term, while significantly relaxing the conventional assumptions typically applied to the relaxation function. Our results extend and improve upon existing findings in the literature, providing deeper insights into the dynamics and stability of suspension bridge models.