Free vibration analysis of 3D non-uniform beam: the rigid segment approach

Abstract

© 2020 Elsevier Ltd This paper presents the free vibration analysis of 3D non-uniform Euler-Bernoulli and Timoshenko beams by using the rigid segment method. In the first step, the rigid segment model of the free-free beam with the constant cross section was presented. In this model the elastic beam was discretized by three rigid segments which are connected by elastic joints with six degrees of freedom. The absolute coordinates of the rigid segments in accordance to the inertial frame were used which further led to the appearance of redundant coordinates. For this reason, the Lagrange equations of the first kind was used to describe the motion of rigid segments. After elimination of the Lagrange multipliers, differential equations of motion expressed in independent coordinates were obtained. Finally, the non-uniform beam is approximately described with the number of uniform beams. The differential equations of the whole non-uniform beam were obtained by summing all of the previously obtained differential equations of the uniform beams. The precision of the proposed algorithm is verified by numerical examples.