closed-form analysis of free vibration of a two-span beam in the frequency domain

 the purpose of this study is to deal with the free vibration of a two-span euler-bernoulli beam. the ratios of flexural rigidity and mass per unit length of two spans of beam are different and variable. the governing eigenvalue problem of system, including two partial differential equations and eight boundary conditions, is introduced and response of system by means of fourier transform is obtained in the frequency domain. an exact closed-form solution is presented and frequency equation of beam is achieved. this equation is numerically solved and expressions for natural frequencies and mode shapes of beam are provided. furthermore, the sensitivity analysis is carried out based on the variation of the right span geometrical properties. in fact, the effect of flexural rigidity and mass per unit length on the natural frequencies of the whole system is investigated and also this question is answered that adding a new span to a single-span simple beam, and increasing or decreasing the flexural rigidity and mass per unit length of it, how can affect the natural frequencies of the two-span beam?