Abstract
This paper presents a comprehensive analytical study of the transverse vibration of a simply supported Euler–Bernoulli beam with two intermediate concentrated masses. Two cases are considered: when the masses are identical and non-identical, and when they are located symmetrically and asymmetrically relative to the middle of the beam. The main objective is to derive closed-form approximate expressions for the first two natural frequencies, which are particularly relevant in engineering applications involving structural dynamics, sensor systems and vibration analysis. The problem is modelled using a characteristic determinant obtained from the exact solution of the fourth-order differential equation of motion, with continuity and jump conditions imposed at the locations of the masses. The obtained closed formulas are used to calculate the natural frequencies asymptotically for small deviations in the values and location of the concentrated masses. A comparative analysis is carried out between the asymptotic and exact values of the natural frequencies.