Exact solution for free vibration of elastically restrained cantilever non-uniform beams joined by a mass-spring system at the free end

Abstract

This article focuses on the free vibration analysis of a non-uniform cantilever beam with an attached mass-spring system at the free end. One end of the beam is elastically restrained against rotation and translation. The height of the beam is assumed constant but the width of the beam exponentially varies. The governing differential equations of the beam, which is a partial differential equation with variable coefficients, and that of the mass-spring system, which is an ordinary differential equation, are found. The exact solution of the problem is then obtained using the pertinent boundary conditions. The eigenvalues and eigenfunctions of the problem which are frequencies and mode shapes of the system are derived for various properties of the system such as stiffness of springs and attached mass. Some limiting cases with available results in the literature are analyzed. The comparison of the proposed results and the reference results shows the accuracy of the solution.