Dynamic basic displacement functions for free vibration analysis of tapered beams

Abstract

Introducing new functions, namely Basic Displacement Functions (BDFs), free transverse vibration of non-prismatic beams is studied from a mechanical point of view. Following structural/mechanical principles, new dynamic shape functions could be developed in terms of BDFs. The new shape functions appear to be dependent on the circular frequency, configuration of the element and its physical properties such as mass density and modulus of elasticity. Differential transform method, a relatively new efficient semi analytical-numerical tool for solving differential equations, is employed to obtain BDFs via solving the governing differential equation for transverse motion of non-prismatic beams. The method poses no restrictions on either type of cross-section or variation of cross-sectional dimensions along the beam element. Using the present element, free vibration analysis is carried out for five numerical examples including beams with (a) linear mass and inertia, (b) linear mass and fourth order inertia, (c) second order mass and fourth order inertia, (d) tapered beam with non-classical boundary conditions and (e) exponentially varying area and inertia. It is observed that the results are in good agreement with the previously published ones in the literature. Finally convergence of the method is investigated.