Another Approach for Obtaining the Eigenvalues of a Bernoulli-Euler Beam with a Single In-span Elastic Rod with a Tip Mass

Abstract

This paper presents a different approach for obtaining the eigenvalues of a combined system investigated previously. The mechanical vibration system in question consists of a horizontal Bernoulli-Euler beam carrying an axially vibrating rod with a tip mass at an intermediate location. In the context of the application of the Galerkin’s procedure and assumed modes approach, the eigenfunctions of the fixed-free rod were previously used as trial functions in a study from the same authors [1]. As an alternative approach, the eigenfunctions of the rod with a tip mass are used as trial functions in this study. Using these eigenfunctions causes very fast convergence with a few numbers of trial functions to determine the eigenvalues. This fact is clearly demonstrated by means of tables and graphs as examples of Bernoulli-Euler beams with different boundary conditions. Also, the numerical results are compared with the values found by the finite element method (FEM), which can be considered as the exact eigenfrequencies of the system. This novel strategy accelerates the convergence rates. The efficacy of the method is demonstrated numerically. In addition, this study successfully demonstrates for the first time that when a continuous system is attached to the horizontal beam, the combined system has new eigenvalues between the original pair of the beam frequencies nearest to the eigenvalues of the attached continuous system, i.e., the axially vibrating rod with a tip mass.