Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy Analysis

Abstract

This study analyzes the fourth-order nonlinear free vibration of a Timoshenko beam. We discretize the governing differential equation by Galerkin's procedure and then apply the homotopy analysis method (HAM) to the obtained ordinary differential equation of the generalized coordinate. We derive novel analytical solutions for the nonlinear natural frequency and displacement to investigate the effects of rotary inertia, shear deformation, pre-tensile loads and slenderness ratios on the beam. In comparison to results achieved by perturbation techniques, this study demonstrates that a first-order approximation of HAM leads to highly accurate solutions, valid for a wide range of amplitude vibrations, of a highorder strongly nonlinear problem.

Nonlinear Vibrations in Homogeneous Nonprismatic Timoshenko Cantilevers

“…Numerous advanced numerical methods are applied to study the nonlinear vibrations of a Timoshenko beam. For instance, Shahlaei-Far et al [26] studied the fourth-order nonlinear free vibrations of cantilever Timoshenko beams using the method of homotopy. They derived analytical expressions for the nonlinear eigenfrequencies and studied the effects of rotatory inertia, shear deformation, and slenderness ratio on the response.…”

“…Note that expression (26) follows the approximation of spatial derivative at the last grid node by the right-hand discrete operator of the second order.…”

“…Natural vibrations of the beam are described by equations ( 23) and ( 24) when ( , ) = 0 and = 0 and boundary conditions are as equations ( 25) and (26). These equations are a finite-difference approximation of the mixed initial-boundary-value problem defined by equations ( 9), ( 10), (13), and ( 14), which depends on the spatial resolution of the grid N G .…”

The Equivalent Linearization Method with a Weighted Averaging for Analyzing of Nonlinear Vibrating Systems

“…In recent years, many powerful methods are used to find approximate solution as well as the amplitude- frequency relationship to the nonlinear differential equations. Some of these methods are Homotopy Perturbation Method (HPM) (He, 1999;He, 2004a;He, 2004b;He, 2004c;Turgut et al, 2007;Bayat et al, 2012), Max-Min Approach (MMA) (He, 2008;Chen et al, 2011;Dumaz et al, 2011;Yazdi et al, 2012;Bayat et al, 2012), Variational Iteration Method (VIM) (Bayat et al, 2012), Energy Balance Method (EBM) (Ganji et al, 2009;Khah et al, 2010;Younesian et al, 2010;Bayat et al, 2012), Amplitude-Frequency Formulation (AFF) (Chen et al, 2011;Jouyburi et al, 2014;Bayat et al, 2012), Parameter Expansion Method (PEM) (Kayaa et al, 2009;Dumaz et al, 2011;Darvishia et al, 2008;Zhao, 2009;Bayat et al, 2012 ), Homotopy Analysis Method (HAM) (He, 2004c;Bayat et al, 2012, Shahram Shahlaei-Far et al, 2016, Modified Homotopy Perturbation Method (MHPM) (Jouybari et al, 2014), Equivalent linearization Method (ELM) (Krylov et al, 1943;Caughey, 1959;Iyengar, 1988;Anh et al, 1995;Anh et al, 1997;Elishakoff et al, 2009;Anh, 2015) and combining Newton's Method with the Harmonic Balance Method (Lim et al, 2006).…”

An extended harmonic balance method based on incremental nonlinear control parameters

“…The Max-min [6], the parameter-expanding approach [7], frequency-amplitude formulation [8], the Variational Iteration [9], perturbation techniques [10,11,12], the iteration perturbation [13], the Homotopy Analysis [14,15], the Energy Balance analysis [16], the harmonic balance [17], the equivalent linearization method (ELM) [18,19] and 20 the Extended Lindstedt-Poincare approach [20] are some examples of these techniques. Each of these methods has some strong points and some weakness.…”