Skip to main content

Scrutiny of non-linear differential equations Euler-Bernoulli beam with large rotational deviation by AGM

Abstract

The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative constitutive equations can allow for viscoelastic or plastic beam deformation. Euler-Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams and geometrically nonlinear beam deflection. In this study, solving the nonlinear differential equation governing the calculation of the large rotation deviation of the beam (or column) has been discussed. Previously to calculate the rotational deviation of the beam, the assumption is made that the angular deviation of the beam is small. By considering the small slope in the linearization of the governing differential equation, the solving is easy. The result of this simplification in some cases will lead to an excessive error. In this paper nonlinear differential equations governing on this system are solved analytically by Akbari-Ganji’s method (AGM). Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration solving. The solution of the problem can be obtained very simply and easily. Furthermore, to enhance the accuracy of the results, the Taylor expansion is not needed in most cases via AGM manner. Also, comparisons are made between AGM and numerical method (Runge-Kutta 4th). The results reveal that this method is very effective and simple, and can be applied for other nonlinear problems.

This is a preview of subscription content, access via your institution.

References

  1. Schmidt A P, Sidebottom R J. Advanced Mechanics of Materials. New York: John Wiley & Sons, Inc, 1993

    Google Scholar 

  2. Libai A, Simmonds J G. The Nonlinear Theory of Elastic Shells. Cambridge: Cambridge University Press, 1998

    Book  MATH  Google Scholar 

  3. Parcel J I, Moorman R B B. Analysis of Statically Indeterminate Structures. New York: JohnWiley & Sons, Inc, 1955

    Google Scholar 

  4. Rostami A K, Akbari M R, Ganji D D, et al. Investigating Jeffery-Hamel flow with high magnetic field and nanoparticle by HPM and AGM. Central European Journal of Engineering, 2014, 4(4): 357–370

    Article  Google Scholar 

  5. Akbari M R, Ganji D D, Majidian A, Ahmadi A R. Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM. Frontiers of Mechanical Engineering, 2014, 9(2): 177–190

    Article  Google Scholar 

  6. Ganji D D, Akbari M R, Goltabar A R. Dynamic vibration analysis for non-linear partial differential equation of the beam-columns with shear deformation and rotary inertia by AGM. Development and Applications of Oceanic Engineering (DAOE), 2014, 3: 22–31

    Google Scholar 

  7. Akbari MR, Ganji D D, Ahmadi A R, et al. Analyzing the nonlinear vibrational wave differential equation for the simplified model of tower cranes by algebraic method. Frontiers of Mechanical Engineering, 2014, 9(1): 58–70

    Article  Google Scholar 

  8. Akbari M R, Ganji D D, Nimafar M, et al. Significant progress in solution of nonlinear equations at displacement of structure and heat transfer extended surface by new AGM approach. Frontiers of Mechanical Engineering, 2014 (in press)

    Google Scholar 

  9. Salimi M. Passive and active control of structures. Dissertation for the Master Degree. Tehran: Tarbiat Modarres University, 2003

    Google Scholar 

  10. Yang J N, Danielians A, Liu S C. A seismic hybrid control systems for building structures. Journal of Engineering Mechanics, 1991, 117(4): 836–853

    Article  Google Scholar 

  11. Ganji Z Z, Ganji D D, Asgari A. Finding general and explicit solutions of high nonlinear equations by the exp-function method. Computers & Mathematics with Applications (Oxford, England), 2009, 58(11–12): 2124–2130

    Article  MATH  MathSciNet  Google Scholar 

  12. Ganji D D. The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer. Physics Letters [Part A], 2006, 355(4–5): 337–341

    Article  MATH  MathSciNet  Google Scholar 

  13. Ning J G, Wang C, Ma T B. Numerical analysis of the shaped charged jet with large cone angle. International Journal of Nonlinear Science and Numerical Simulation, 2006, 7(1): 71–78

    Article  MathSciNet  Google Scholar 

  14. Das S, Gupta P. Application of homotopy perturbation method and homotopy analysis method to fractional vibration equation. International Journal of Computer Mathematics, 2011, 88(2): 430–441

    MATH  MathSciNet  Google Scholar 

  15. Ghosh S, Roy A, Roy D. An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators. Computer Methods in Applied Mechanics and Engineering, 2007, 196(4–6): 1133–1153

    Article  MATH  MathSciNet  Google Scholar 

  16. Sheikholeslami M, Ganji D D, Ashorynejad H R, et al. Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method. Applied Mathematics and Mechanics, 2012, 33(1): 25–36

    Article  MATH  MathSciNet  Google Scholar 

  17. Sfahani M G, Barari A, Omidvar M, et al. Dynamic response of inextensible beams by improved energy balance method. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2011, 225(1): 66–73

    Google Scholar 

  18. Ganji Z Z, Ganji D D, Bararnia H. Approximate general and explicit solutions of nonlinear BBMB equations exp-function method. Applied Mathematical Modelling, 2009, 33(4): 1836–1841

    Article  MATH  MathSciNet  Google Scholar 

  19. Ren Z F, Liu G Q, Kang Y X, et al. Application of He’s amplitude—application of He’s amplitude frequency formulation to nonlinerar oscillators with discontinuities. Physica Scripta, 2009, 80(4): 45003

    Article  Google Scholar 

  20. He J H. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 2006, 20(10): 1141–1199

    Article  MATH  MathSciNet  Google Scholar 

  21. Wu X H, He J H. Solitary solutions, periodic solutions and compacton-like solutions using the exp-function method. Computers & Mathematics with Applications, 2007, 54(7–8): 966–986

    Article  MATH  MathSciNet  Google Scholar 

  22. He J H. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 1999, 178(3–4): 257–262

    Article  MATH  MathSciNet  Google Scholar 

  23. Zhou J K. Differential Transformation and Its Applications for Electrical Circuits. Wuhan: Huazhong University Press, 1986

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. R. Akbari.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Akbari, M.R., Nimafar, M., Ganji, D.D. et al. Scrutiny of non-linear differential equations Euler-Bernoulli beam with large rotational deviation by AGM. Front. Mech. Eng. 9, 402–408 (2014). https://doi.org/10.1007/s11465-014-0316-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11465-014-0316-8

Keywords

  • AGM
  • critical load of columns
  • large deformations of beam
  • nonlinear differential equation