Application of Second-Order Boundary Conditions in the Vibrations of Beams With Attached Lumped Mass Under Axial Force

Abstract

This paper addresses the exact solution of a beam’s free vibrations with a concentrated mass within its intervals when the beam undergoes an axial loading. The Euler-Bernoulli beam theory is used as the basis of the model of the beam. This problem has been extensively studied before in the literature. The main contribution of this paper is in its novel approach to treating the effect of the concentrated mass and solving the resulting governing equation of motion. The effect of the concentrated mass is incorporated into the governing partial differential equation (PDE), rather than being treated as a boundary condition. To this end, the effect of a distributed transverse force, a distributed moment, and axial loading is included in the governing PDE. Then, the properties of Dirac’s delta function is used to represent the attached concentrated mass as a displacement-dependent concentrated transverse force. As a results, the PDE has the delta function as a coefficient of one of its terms. This type of equations are traditionally solved in frequency domain using Laplace transforms. In this paper, the technique to solve this PDE in time domain is presented. It has been demonstrated that this technique intrinsically leads to the application of the second-order theory. The exact natural frequencies and mode shapes of vibration of the system are determined by solving the PDE for free vibrations.