The results presented in this chapter are largely inspired by our work published in (Aissa and Ahmedi, SeMA (2025). https://doi.org/10.1007/s40324-025-00384-w ). Building upon these findings, we investigate the stabilization of a system of locally coupled wave and Euler-Bernoulli beam equations with local Kelvin-Voigt damping. Our objective is to provide a comprehensive analysis of the system’s stability properties, considering different geometric and structural configurations of the damping and coupling regions. More precisely, we examine three distinct cases to assess the impact of localized dissipation on the long-term behavior of the system. First, we explore the scenario where the supports of the damping and coupling coefficients are disjoint, leading to a more challenging stability analysis due to the lack of direct interaction between the damping and coupling effects. Then, we shift our focus to cases where the damping and coupling regions overlap, which allow us to evaluate how their interaction influences energy dissipation and decay rates. Through this study, we aim to characterize the qualitative properties of the system, establish sharp decay estimates, and highlight the limitations of exponential stability under global Kelvin-Voigt damping. The results obtained extend and refine existing stability criteria for coupled wave-beam systems, contributing to a deeper understanding of their asymptotic behavior.