Analytical Approximation of Nonlinear Vibration of Euler-Bernoulli Beams

Abstract

In this paper, the Homotopy Analysis Method (HAM) with two auxiliary parameters and Differential Transform Method (DTM) are employed to solve the geometric nonlinear vibration of EulerBernoulli beams subjected to axial loads. A second auxiliary parameter is applied to the HAM to improve convergence in nonlinear systems with large deformations. The results from HAM and DTM are compared with another popular numerical method, the shooting method, to validate these two analytical methods. HAM and DTM show excellent agreement with numerical results (the maximum errors in our calculations are about 0.002%), and they additionally provide a simple way to conduct a parametric analysis with different physical parameters in Euler-Bernoulli beams. To show the benefits of this method, the effect of different physical parameters on the amplitude is discussed for a cantilever beam with a cyclically varying axial load.

A New Beam Model for Simulation of the Mechanical Behaviour of Variable Thickness Functionally Graded Material Beams Based on Modified First Order Shear Deformation Theory

“…The nonlinear bending of a two-directionally FGM beam was considered by Li et al [5] using the generalized differential quadrature method (GDQM). Jafari et al [6] used analytical approximation for nonlinear vibration analysis of Euler-Bernoulli beams. Alshorbagy et al [7] used FEM to analyze free vibration of a Euler-Bernoulli beam made of FGM.…”

An Explicit Solution to Continuum Compliant Cantilever Beam Problem with Various Variational Iteration Algorithms

“…However, the laborious computational involvement of these methods has deterred researchers from using them. Recently, some approximate analytical solutions are proposed, such as, Adomian decomposition method (ADM) [11][13] [14], homotopy analysis method (HAM) [4][15] [16], homotopy perturbation method (HPM) [15], [17]- [22], Chebyshev's polynomial approximation method [23] and differential transform method (DTM) [24]- [27]. The present study aim at exploring the various algorithms of variational iteration method (VIM) in the solution the nonlinear differential equation resulting from the large deformation of a continuum compliant cantilever beam under point load at the free.…”

Flow and heat transfer analysis of magnetohydrodynamic (MHD) second-grade fluid in a channel with a porous wall

“…The HAM technique has been used successfully by Ali et al [19][20][21][22] in their recent investigations on the flow and heat transfer of a non-Newtonian fluid over an oscillatory stretching sheet. Jafari et al [23] have recently employed HAM technique to solve the Abstract Steady flow of a magnetohydrodynamic (MHD) second-grade fluid in a parallel channel is considered. One of the heated walls is cooled by coolant injection through the other porous wall.…”