Exact Closed-Form Solution for Vibration Analysis of Beams Carrying Lumped Masses with Rotary Inertias

In this paper, an exact closed-form solution is presented for free vibration analysis of Bernoulli–Euler beams carrying attached masses with rotary inertias. The proposed technique explicitly provides frequency equation and corresponding mode as functions with two integration constants which should be determined by external boundary conditions implementation and leads to the solution to a two by two eigenvalue problem. The concentrated masses and their rotary inertia are modeled using Dirac’s delta generalized functions without implementation of continuity conditions. The nondimensional inhomogeneous differential equation of motion is solved by applying integration procedure. Using the fundamental solutions which are made of the appropriate linear composition of trigonometric and hyperbolic functions leads to making the implementation of boundary conditions much easier. The proposed technique is employed to study the effects of quantity, position and translational and rotational inertia of the concentrated masses on the dynamic behavior of the beam for all standard boundary conditions. Unlike many of the previous exact approaches, the presented solution has no limitation in a number of concentrated masses.