Dynamic response of a beam carrying a lumped mass along its span

In this paper, we study the periodic solutions for the free vibration of a conservative oscillator with fifth-order nonlinearity using two new methods, the energy balance method and the variational approach. We compare the results obtained with those yielded by the traditional harmonic balance method and equivalent linearization–nonlinearization method. The periodic solutions are then verified using the Runge–Kutta approach.

[1]  A. Dimarogonas.  Vibration for engineers , 1992 .

[2]  Davood Domiri Ganji,et al.  Application of He's energy balance method to Duffing-harmonic oscillators , 2011, Int. J. Comput. Math..

[3]  H. Mirgolbabaei,et al.  Analytical solution of forced-convective boundary-layer flow over a flat plate , 2010 .

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

[5]  Nobukatsu Okuizumi,et al.  Multiple time scale analysis of hysteretic systems subjected to harmonic excitation , 2004 .

[6]  G. Failla,et al.  A solution method for Euler–Bernoulli vibrating discontinuous beams , 2008 .

[7]  L. Meirovitch,et al.  Fundamentals of Vibrations , 2000 .

[8]  Ji-Huan He.  Some new approaches to Duffing equation with strongly and high order nonlinearity (II) parametrized perturbation technique , 1999 .

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

[10]  J. Herder,et al.  Large deflection analysis of cantilever beam under end point and distributed loads , 2014 .

[11]  S. S. Ganji,et al.  HES ENERGY BALANCE AND HES VARIATIONAL METHODS FOR NONLINEAR OSCILLATIONS IN ENGINEERING , 2009 .

[12]  Lars Bo Ibsen,et al.  Analysis of highly nonlinear oscillation systems using He’s max-min method and comparison with homotopy analysis and energy balance methods , 2010 .

[13]  Ji-Huan He.  Homotopy Perturbation Method for Bifurcation of Nonlinear Problems , 2005 .

[14]  S. K. Lai,et al.  Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators , 2009 .

[15]  M. G. Sfahani,et al.  Higher-Order Approximation of Cubic-Quintic Duffing Model , 2011 .

[16]  Amin Barari,et al.  DYNAMIC RESPONSE OF AXIALLY LOADED EULER-BERNOULLI BEAMS , 2011 .

[17]  S. Caddemi,et al.  Closed form solutions of Euler–Bernoulli beams with singularities , 2005 .

[18]  Amin Barari,et al.  Analytical Solutions to Non-linear Mechanical Oscillation Problems , 2011 .

[19]  M. G. Sfahani,et al.  Consideration of transient stream/aquifer interaction with the nonlinear Boussinesq equation using HPM , 2011 .

[20]  Amin Barari,et al.  Analytical solutions to nonlinear conservative oscillator with fifth-order nonlinearity , 2010 .

[21]  Lars Bo Ibsen,et al.  Nonlinear vibration of an elastically restrained tapered beam , 2012 .

[22]  J. Ramos.  An artificial parameter–Linstedt–Poincaré method for oscillators with smooth odd nonlinearities , 2009 .

[23]  D. Ganji,et al.  Assessment of water penetration problem in unsaturated soils , 2009 .

[24]  M. N. Hamdan,et al.  Large Amplitude Free Vibrations of a Uniform Cantilever Beam Carrying AN Intermediate Lumped Mass and Rotary Inertia , 1997 .

[25]  Erik Lund,et al.  On Approximate Analytical Solutions of Nonlinear Vibrations of Inextensible Beams Using Parameter-Expansion Method , 2010 .

[26]  Davood Domiri Ganji,et al.  Approximate analysis of two-mass-spring systems and buckling of a column , 2011, Comput. Math. Appl..

[27]  Cha'o-Kuang Chen,et al.  Application of the differential transformation method to the free vibrations of strongly non-linear oscillators , 2009 .

[28]  M. N. Hamdan,et al.  ON THE LARGE AMPLITUDE FREE VIBRATIONS OF A RESTRAINED UNIFORM BEAM CARRYING AN INTERMEDIATE LUMPED MASS , 1997 .

[29]  D. Ganji,et al.  Application of He’s Variational Iteration Method for Solving Nonlinear BBMB Equations and Free Vibration of Systems , 2009 .

[30]  M. Hamdan,et al.  BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS , 2002 .

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

[32]  Ji-Huan He,et al.  Recent development of the homotopy perturbation method , 2008 .

[33]  M. N. Hamdan,et al.  Nonlinear vibrations and buckling of a flexible rotating beam: A prescribed torque approach , 2007 .

[34]  Hans Peter Gottlieb,et al.  Harmonic balance approach to limit cycles for nonlinear jerk equations , 2006 .

[35]  Weichung Yeih,et al.  Applications of dual MRM for determining the natural frequencies and natural modes of an Euler-Bernoulli beam using the singular value decomposition method , 1999 .

[36]  R. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer. By D. A ANDERSON, J. C. TANNEHILL and R. H. PLETCHER. Hemisphere, 1984. 599 pp. $39.95. , 1986, Journal of Fluid Mechanics.

[37]  Ji-Huan He.  AN ELEMENTARY INTRODUCTION TO RECENTLY DEVELOPED ASYMPTOTIC METHODS AND NANOMECHANICS IN TEXTILE ENGINEERING , 2008 .

[38]  Ji-Huan He,et al.  Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant , 2002 .

[39]  G. Tomlinson,et al.  Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis , 2008 .

[40]  S. S. Ganji,et al.  Analytical Evaluation of Jamming Transition Problem , 2011 .

[41]  D. Ganji,et al.  An Approximate Solution for Boundary Value Problems in Structural Engineering and Fluid Mechanics , 2008 .

[42]  D. Zwillinger.  Handbook of differential equations , 1990 .

[43]  Anindya Chatterjee,et al.  Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales , 2006 .

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

[45]  Fawang Liu,et al.  Finite difference approximations for the fractional Fokker–Planck equation , 2009 .

[46]  A. Barari,et al.  Highly Nonlinear Temperature-Dependent Fin Analysis by Variational Iteration Method , 2010 .

[47]  J. H. Tang,et al.  Solution of a Duffing-harmonic oscillator by the method of harmonic balance , 2006 .

[48]  Nilanjan Coomar,et al.  Comparative analysis of steady state heat transfer in a TBC and functionally graded air cooled gas turbine blade , 2010 .

[49]  Ding Zhou,et al.  Dynamic characteristics of a beam and distributed spring-mass system , 2006 .

[50]  Ji-Huan He,et al.  Application of Parameter-expanding Method to Strongly Nonlinear Oscillators , 2007 .

[51]  Ji-Huan He,et al.  Nonlinear oscillator with discontinuity by parameter-expansion method , 2008 .

[52]  Osman Kopmaz,et al.  On the Curvature of an Euler–Bernoulli Beam , 2003 .

[53]  J. Liu,et al.  A new method based on the harmonic balance method for nonlinear oscillators , 2007 .

[54]  F-C Wang,et al.  Designing and testing a hydraulic inerter , 2011 .

[55]  Amin Barari,et al.  Dynamic Response of Inextensible Beams by Improved Energy Balance Method , 2011 .

[56]  Davood Domiri Ganji,et al.  Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations , 2008 .

[57]  S. S. Ganji,et al.  Differential transform method for mathematical modeling of jamming transition problem in traffic congestion flow , 2012, Central Eur. J. Oper. Res..