This study focuses on the free bending vibrations of two isotropic homogeneous Euler-Bernoulli beams coupled elastically by an elastic double spring-mass system. The upper beam is supported by two point masses at arbitrary positions, taking into account their rotational inertia. The main objective is to determine the natural frequencies of free vibration and to characterize the associated vibration modes for this complex structure. The methodology adopted consists of dividing the structure into portion of beam and then applying boundary and continuity conditions to form a matrix system. The matrix system thus formed yields two solutions, the non-trivial solution of which is obtained by solving the nonlinear equation using the Newton-Raphson iterative method. The validation of the results obtained with those found in the literature is carried out in the following way: we compare the results of the free vibration of the beam supporting two masses, then we compare the results of the free vibration of two beams coupled by a double mass spring in order to obtain our own results. This research will enrich the literature with new ideas and serve as a reference for future research in this field.