Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams

Presenting new functions, basic displacement functions (BDFs), a novel method based on mechanical/structural principles is introduced for free vibration analysis of arbitrarily tapered Timoshenko beams in preference to primarily mathematically based methodologies. BDFs are obtained through solving the governing equations of transverse motion of Timoshenko beams by means of power series method. It is shown that exact shape functions and consequently structural matrices could be derived in terms of BDFs through a purely mechanical approach. It is also demonstrated that the new shape functions depend on the geometrical and mechanical properties of the beam element and moreover, contrary to usual shape functions used in FE methods, are susceptible to the circular frequency. In order to verify the competency of the proposed method, several numerical examples for classical as well as non-classical boundary conditions are carried out and the results are compared with those in the literature.