Computational model of vibration of Timoshenko beam under axial loads

The free vibration characteristics of a cantilever tapered Timoshenko beam is analyzed in this study. First, the strain displacement relationship for the Timoshenko beam is formulated and used to derive the kinetic and strain energies in explicit analytical form. Second, Lagrange’s variational principle is used to derive the governing differential equation of motion and the associated boundary conditions. Third, the Rayleigh-Ritz Method (R-R) is applied to the equation of motion and the boundary conditions to form a set of algebraic equations from which the frequency equation is derived. Next, a numerical algorithm implemented in the software package Mathematica is used to compute the natural frequencies. Also, the variation of the natural frequencies of vibration with respect to variations in the taper ratio and also the slenderness ratio is studied. The results obtained from the Timoshenko theory are compared with results obtained in literature to demonstrate the accuracy and relevance of the their application.