Free vibration of axially functionally graded Timoshenko cantilever beam with a large rigid body attached at its free end

Abstract

The problem of natural frequency computing of an axially functionally graded (AFG) Timoshenko cantilever beam of a constant cross-sectional area is considered. The longitudinal and transverse vibrations of the cantilever beam are coupled because of the boundary conditions at its right end, where the eccentrically displaced rigid body is attached. Once the variables are separated, the procedure of solving the system of partial differential equations is reduced to solving the two-point boundary value problem of ordinary linear differential equations with nonlinear coefficients and linear boundary conditions. In this case, it is possible to transfer boundary conditions and to reduce the problem to the Cauchy problem of initial conditions. Also, it is possible to analyze the influence of various parameters on the values of natural frequencies. The impact of the cross-sectional area on natural frequency is analyzed and illustrated by a numerical example.