Analytical study on the vibration frequencies of tapered beams

Abstract

A vast amount of published work can be found in the field of beam vibrations dealing with analytical and numerical techniques. This paper deals with analysis of the nonlinear free vibrations of beams. The problem considered represents the governing equation of the nonlinear, large amplitude free vibrations of tapered beams. A new implementation of the ancient Chinese method called the Max-Min Approach (MMA) and Homotopy Perturbation Method (HPM) are presented to obtain natural frequency and corresponding displacement of tapered beams. The effect of vibration amplitude on the non-linear frequency is discussed. In the end to illustrate the effectiveness and convenience of the MMA and HPM, the obtained results are compared with the exact ones and shown in graphs and in tables. Those approaches are very effective and simple and with only one iteration leads to high accuracy of the solutions. It is predicted that those methods can be found wide application in engineering problems, as indicated in this paper. Keywords large amplitude free vibrations, analytical solution, tapered beam, Homotopy Perturbation Method (HPM), Max-Min Approach (MMA).

Dynamic model of large amplitude vibration of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia

“…So, many new techniques have appeared in the open literature such as: variational iteration method (He, 2007a;Barari et al, 2011), homotopy perturbation method Saravi et al, 2013), homotopy analysis method (Liao, 2003;Khan et al, 2012)and some other methods Ghasempoor et al, 2102;Akbarzade and Khan, 2012;Alinia et al, 2011;Ganji, 2012;Torabi et al, 2012;Sheikholeslami et al, 2012a;2012b;Bayat et al, 2011;Rafieipour et al, 2012;Marinca and Herisanu, 2010;Salehi et al, 2012;Hamidi et al, 2012) The variational iteration method (VIM) and variational approach (VA) which were introduced by 2007a;2007b) are very powerful methods in solving non-linear differential equations and studying nonlinear vibration of beams. The Variational Approach which is also called He's variational approach, as well as the variational iteration method, was utilized here to obtain the analytical expression for the following model of nonlinear oscillations in the structural engineering problems:…”

Nonlinear Vibration Analysis of Euler-Bernoulli Beams by Using Continuous Galerkin-Petrov Time-Discretization Method

“…To study their behavior it is therefore important to design methods for the numerical solutions of these partial differential equations. In this respect, recently some approximate analytical techniques (see for instance, Bayat et al, 2010;Baghari et al, 2014;Jafari et al, 2014;Bayat et al, 2011;, Ganji, 2012;He, 1999;He, 2006;Khan et al, 2012;Liu and Gurram, 2009;Mirgolbabaei et al, 2010;Sfahani et al, 2011) as well as numerical methods by Lai et al (2008) (see also, Boukhalfa et al, 2010 Ganji et al, 2011) and references therein, have been proposed to investigate the nonlinear vibrations of beams for designing and fabrication purpose.…”

An Analytical Solution for Free Vibration of Elastically Restrained Timoshenko Beam on an Arbitrary Variable Winkler Foundation and Under Axial Load

“…The double Fourier transform is employed for the free vibration analysis of the semi-rigid connected Reddy-Bickford beam with variable cross-section on elastic soil and under axial load by Yesilce and Catal (Yesilce and Catal, 2011). Bayat et al presented the analytical study on the vibration frequencies of tapered beams (Bayat et al, 2011). The Max-Min Approach and Homotopy Perturbation Method are employed in order to solve the governing equations of tapered beams.…”