This study focuses on the analysis of the transverse vibratory motion of two homogeneous and isotropic beams, clamped at their ends. The lower beam rests on a Winkler-Pasternak foundation and is coupled to the upper beam by two elastic systems, modeled as double spring-mass. The problem is analyzed within the framework of the Euler-Bernoulli beam theory. The main objective of this study is to determine the natural frequencies of free vibration and the vibration modes associated with this complex structural configuration. The results obtained for the elastically coupled system are validated by comparison with those of a simplified configuration featuring a single double spring-mass elastic coupling, as reported in the literature. The adopted methodology is based on subdividing of the structure into beam segments, followed by the application of boundary and compatibility conditions. The system thus formulated is then solved using the iterative Newton-Raphson method, which finds the natural vibration frequencies of the vibrations and, consequently, the associated vibration modes. This approach provides an accurate analysis of the vibration phenomena within the dynamic modeling of the structure.