A new numeric–symbolic procedure for variational iteration method with application to the free vibration of generalized multi-span Timoshenko beam

Abstract

In this article, a novel approach is introduced for the free vibration analysis of beams based upon the variational iteration method. The new approach uses a numeric–symbolic procedure that tackles the problem of increased execution time involved in symbolic integrations. This drawback is usually encountered in solving complicated free vibration problems such as stepped beams connected to lumped parameter subsystems. The proposed procedure is applied for free vibration analysis of a generalized multi-span Timoshenko beam connected to multiple lumped subsystems. Each subsystem is represented by a two-degree-of-freedom spring–mass–damper system. Several verification examples are presented where the results of the proposed numeric–symbolic variational iteration method are compared with the conventional symbolic approach symbolic variational iteration method in terms of execution time. Special attention is given to the verification of the new results against finite element modeling results and exact solutions where possible. Based on the presented results, it is shown that the new numeric–symbolic variational iteration method procedure efficiently reduces the time required for solving the free vibration problem while maintaining the high accuracy and robustness of the variational iteration method. The new procedure presented here may facilitate solving some engineering problems in which the conventional symbolic approach usually fails to solve owing to extensive memory requirements. The study contributes toward further improvements of the variational iteration method and its application to sophisticated dynamic systems.

Vibration analysis of cracked beam based on Reddy beam theory by finite element method

“…The significance of shear deformations in elastic beams was first demonstrated by Timoshenko (1921) and Timoshenko beam theory can be considered as a first-order shear deformation theory (FSDT). Timoshenko beam theory is widely used for thick beams (Cheng et al, 2011; El-Sayed and Farghaly, 2018, 2020; El-Sayed and El-Mongy, 2021; Elsawaf et al, 2020; Farghaly and El-Sayed, 2016, 2017; Kaya and Dowling, 2016; Yardimoglu, 2010). In this case, a plane section again remains plane after deformation, as shown in Figure 1(c); however, the assumption that the plane remains perpendicular to the beam axis is relaxed.…”