The exact natural frequency solution of a free axialbending vibration problem of a non-uniform AFG cantilever beam with a tip body

Abstract

In this paper we present the solution of exact equations of motion for coupled axial and bending vibrations of a non-uniform axially functionally graded (AFG) cantilever beam with a body of which mass center is eccentrically displaced in axial and transverse direction with respect to the beam’s end. The Euler-Bernoulli beam theory is implemented to model behavior of the beam under axial and transverse in-plane vibrations. Based on the paper [1], that is supported by results publish in the paper [2] authors confirm the obtained results of natural frequencies of the AFG cantilever beam, when boundary conditions define the vibration coupling. The modified symbolic-numeric method of initial parameters presented in [3] is implemented in computing natural frequencies of the beam. This method is expanded to solve axial-bending vibration problems, with respect to the one presented in the literature [3] for the problem of the vibration of a cantilever beam. Some minor deviations in the obtained results may be noticed with respect to those obtained in [1], yet all within a tolerance domain due to the computational precision.