Efficient modeling of distributed structural systems interacting with elastic media remains a central challenge in computational mechanics. This work presents a semi-analytical spectral framework for the dynamic analysis of free–free Timoshenko beams on Winkler foundation and subjected to transverse ground vibration. Starting from the translational and rotational equilibrium equations, a fourth-order partial differential equation governing transverse motion is derived. Separation of variables reduces the problem to a biquadratic characteristic equation whose discriminant reveals three transition frequencies partitioning the vibration spectrum into distinct regimes. These spectral thresholds determine the qualitative structure of the eigenvalues and generate closed-form modal functions exhibiting different oscillatory regimes. Explicit expressions for natural frequencies and eigenfunctions are obtained under free–free boundary conditions, including rigid-body modes. The forced response under traveling ground excitation is evaluated through modal decomposition and time integration of a reduced-order system, preserving shear deformation, rotary inertia, and soil–structure interaction effects while providing a computationally efficient alternative to full finite element discretization. Building upon recent validated developments in dynamic soil–structure interaction modelling [1], the formulation demonstrates excellent agreement with detailed finite element simulations and available experimental data across the spectral range. The framework captures spectral clustering and modal density growth in long systems, mechanisms that strongly influence dynamic amplification. The identified spectral partitioning establishes a mathematically transparent and computationally robust framework for advanced vibration analysis and provides a systematic foundation for extensions toward nonlocal and fractional-order structural interaction models. REFERENCES [1] G. Banushi, Dynamic analysis of free-free Timoshenko beams on elastic foundation under transverse transient ground deformation. Engineering Structures, 352, 122080, 2026.