Application of the homotopy method for the analytic approach of the nonlinear free vibration analysis of the simple end beams using four engineering theories

摘要

In this paper, nonlinear vibration analyses of Euler-Bernoulli, Rayleigh, Shear and Timoshenko beams with simple end conditions are presented using homotopy analysis method (HAM). Closed form solutions for natural frequencies, beam deflection, post-buckling load-deflection relation, and critical buckling load are presented. The calculated natural frequencies for all four cases were verified against some available results in the literature and very good agreement observed. Furthermore, obtained results for deflection, buckling, and post-buckling of each beam are presented and the effects of some parameters, such as slenderness ratio, the rotary inertia, and the shear deformation are examined.