Analytical study on the vibration frequencies of tapered beams

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.

[1]  M. Sathyamoorthy,et al.  Literature Review : Nonlinear Analysis of Beams Part Ii: Finite Element Methods , 1982 .

[2]  J. H. Lau.  Vibration Frequencies of Tapered Bars With End Mass , 1984 .

[3]  Amin Barari,et al.  Analysis of Blasius Equation for Flat-plate Flow with Infinite Boundary Value , 2010 .

[4]  S. Mohyud-Din,et al.  Parameter-expansion Techniques for Strongly Nonlinear Oscillators , 2009 .

[5]  Elçin Yusufoglu,et al.  An improvement to homotopy perturbation method for solving system of linear equations , 2009, Comput. Math. Appl..

[6]  Davood Domiri Ganji,et al.  New class of solutions for water infiltration problems in unsaturated soils , 2010 .

[7]  Ahmet Yildirim,et al.  Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method , 2009 .

[8]  H. Zhang.  Application of He's Frequency-amplitude Formulation to an x1/3 Force Nonlinear Oscillator , 2008 .

[9]  B. Downs,et al.  Transverse Vibrations of Cantilever Beams Having Unequal Breadth and Depth Tapers , 1977 .

[10]  Ji-Huan He,et al.  Rebuild of King Fang 40 BC musical scales by He's inequality , 2005, Appl. Math. Comput..

[11]  Abdelkrim Boukhalfa,et al.  Free vibration analysis of an embarked rotating composite shaft using the hp-version of the FEM , 2010 .

[12]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[13]  Amin Barari,et al.  The Approximate Analysis of Nonlinear Behavior of Structure under Harmonic Loading , 2010 .

[14]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[15]  D. Evensen.  Nonlinear vibrations of beams with various boundary conditions. , 1968 .

[16]  Modified Variational Iteration Method for Integro-Differential Equations and Coupled Systems , 2010 .

[17]  Lan Xu,et al.  Variational approach to solitons of nonlinear dispersive K(m, n) equations , 2008 .

[18]  Kenzo Sato.  Transverse vibrations of linearly tapered beams with ends restrained elastically against rotation subjected to axial force , 1980 .

[19]  Ji-Huan He,et al.  Max-Min Approach to Nonlinear Oscillators , 2008 .

[20]  Davood Domiri Ganji,et al.  Application of the energy balance method to nonlinear vibrating equations , 2010 .

[21]  G. Venkateswara Rao,et al.  Re-investigation of large-amplitude free vibrations of beams using finite elements , 1990 .

[22]  Larisse Klein,et al.  Transverse vibrations of non-uniform beams , 1974 .

[23]  Davood Domiri Ganji,et al.  Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire , 2010 .

[24]  G. Venkateswara Rao,et al.  Analysis of the nonlinear vibrations of unsymmetrically laminated composite beams , 1991 .

[25]  Amin Barari,et al.  Analytical Evaluation of the Nonlinear Vibration of Coupled Oscillator Systems , 2011 .

[26]  Hongxun Yi,et al.  Some further results on weighted sharing three values and Brosch's theorem , 2009, Comput. Math. Appl..

[27]  Lawrence W. Rehfield,et al.  Nonlinear free vibrations of elastic structures , 1973 .

[28]  A. Beck,et al.  SOLUTION OF THE STOCHASTIC BEAM BENDING PROBLEM BY GALERKIN METHOD AND THE ASKEY-WIENER SCHEME , 2009 .

[29]  D. Ganji,et al.  Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems , 2010 .

[30]  G. Venkateswara Rao,et al.  Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates , 1976 .

[31]  Ahmet Yildirim,et al.  Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method , 2007, Comput. Math. Appl..

[32]  M. I. Qaisi.  Application of the harmonic balance principle to the nonlinear free vibration of beams , 1993 .

[33]  B. Nageswara Rao,et al.  On nonlinear free vibrations of simply supported uniform beams , 1992 .

[34]  Sen-Yung Lee,et al.  Exact Solutions for the Analysis of General Elastically Restrained Nonuniform Beams , 1992 .