Abstract: In order to overcome the difficulties of studying the vibration analysis model of a multi-span beam system under various boundary and coupling conditions, this paper constructs a free vibration analysis model of a multi-span beam system on the basis of the Bernoulli-Euler beam theory. The vibration characteristics of a multi-span beam system under arbitrary boundary supports and elastic coupling conditions are investigated using the current analysis model. Unlike most existing techniques, the beam displacement function is generally sought as an improved Fourier cosine series, and four sine terms are introduced to overcome all the relevant discontinuities or jumps of elastic boundary conditions. On this basis, the unknown series coefficients of the displacement function are treated as the generalized coordinates and solved using the Rayleigh-Ritz method, and the vibration problem of multi-span bean systems is converted into a standard eigenvalue problem concerning the unknown displacement expansion coefficient. By comparing the free vibration characteristics of the proposed method with those of the FEA method, the efficiency and accuracy of the present method are validated, providing a reliable and theoretical basis for multi-span beam system structure in engineering applications.