A computational dynamics and damping–induced transitions in beam–foundation systems with viscoelastic constraints

This study examines the vibrational dynamics of elastically constrained beam foundation systems by considering Timoshenko, shear, Rayleigh, and Euler–Bernoulli beam models resting on viscoelastic Filonenko-Borodich and Hetényi foundations. Emphasis is placed on understanding how damping, foundation stiffness, shear deformation, and rotary inertia govern dynamic transitions across undamped, underdamped, and overdamped regimes. Using separation of variables for the analytical solution and the Galerkin finite element method for the numerical solution, non–classical boundary conditions are used for the formulation of the equations. Analytical and numerical solutions are employed to explore frequency characteristics, oscillation amplitudes, and stability behavior. The results reveal that increasing foundation stiffness enhances natural frequencies and contributes to effective vibration control, while increasing damping induces a gradual transition from sustained oscillations to complete vibration suppression. These damping-driven transitions highlight complex dynamic behavior relevant to nonlinear systems theory. Excellent agreement between analytical and numerical findings confirms the robustness of the proposed formulation. The study underscores the critical role of viscoelastic foundations and damping mechanisms in shaping nonlinear vibration responses, offering insights applicable to the design of advanced structural systems. Extensions to non–uniform geometries, heterogeneous materials, and environmental effects are expected to further enrich the dynamical behavior and practical relevance of the model.