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Analyzing the nonlinear vibrational wave differential equation for the simplified model of Tower Cranes by Algebraic Method

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Abstract

In the current paper, a simplified model of Tower Cranes has been presented in order to investigate and analyze the nonlinear differential equation governing on the presented system in three different cases by Algebraic Method (AGM). Comparisons have been made between AGM and Numerical Solution, and these results have been indicated that this approach is very efficient and easy so it can be applied for other nonlinear equations. It is citable that there are some valuable advantages in this way of solving differential equations and also the answer of various sets of complicated differential equations can be achieved in this manner which in the other methods, so far, they have not had acceptable solutions. The simplification of the solution procedure in Algebraic Method and its application for solving a wide variety of differential equations not only in Vibrations but also in different fields of study such as fluid mechanics, chemical engineering, etc. make AGM be a powerful and useful role model for researchers in order to solve complicated nonlinear differential equations.

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References

  1. He J H. A note on delta-perturbation expansion method. Applied Mathematics and Mechanics, 2002, 23(6): 634–638

    Article  MATH  Google Scholar 

  2. He J H. Bookkeeping parameter in perturbation methods. International Journal of Nonlinear Sciences and Numerical Simulation, 2001, 2(3): 257

    Article  MATH  MathSciNet  Google Scholar 

  3. Adomian G. Solving Frontier Problems of Physics: the Composition Method. Kluwer, Boston, 1994

    Book  Google Scholar 

  4. Ramos J I. Piecewise-adaptive decomposition methods. Chaos, Solitons, and Fractals, 2008, 198(1): 92

    MATH  Google Scholar 

  5. 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 

  6. Ramos J I. An artificial parameter LinstedtePoincaré method for oscillators with smooth odd nonlinearities. Chaos, Solitons, and Fractals, 2009, 41(1): 380–393

    Article  MATH  Google Scholar 

  7. 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 

  8. Wang S Q, He J H. Nonlinear oscillator with discontinuity by parameter expansion method. Chaos, Solitons, and Fractals, 2008, 35(4): 688–691

    Article  MATH  Google Scholar 

  9. Shou D H, He J H. Application of parameter-expanding method to strongly nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation, 2007, 8(1): 121

    Article  Google Scholar 

  10. Xu L, He J H. He’s parameter-expanding method for strongly nonlinear oscillators. Journal of Computational and Applied Mathematics, 2007, 207(1): 148–154

    Article  MATH  MathSciNet  Google Scholar 

  11. He J H. The homotopy perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics and Computation, 2004, 151(1): 287–292

    Article  MATH  MathSciNet  Google Scholar 

  12. He J H. Approximate analytical solution of Blasiu’s equation. Communications in Nonlinear Science and Numerical Simulation, 1998, 3(4): 260–263

    Article  MATH  Google Scholar 

  13. He J H. Variational iteration method-a kind of non-linear analytical technique: some examples. International Journal of Non-linear Mechanics, 1999, 34(4): 699–708

    Article  MATH  Google Scholar 

  14. He J H, Wan Y Q, Guo Q. An iteration formulation for normalized diode characteristics. International Journal of Circuit Theory and Applications, 2004, 32(6): 629–632

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Rafei M, Ganji D D, Daniali H, Pashaei H. The variational iteration method for nonlinear oscillators with discontinuities. Journal of Sound and Vibration, 2007, 305(4–5): 614–620

    Article  MATH  MathSciNet  Google Scholar 

  17. Clough R W, Penzien J. Dynamics of structures. Third Edition. Computers & Structures. Inc. University Ave. Berkeley, CA 94704, 2003

    Google Scholar 

  18. Chopra A K. Dynamics of Structures: Theory and Applications to Earthquake Engineering. Civil Engineering and Engineering Mechanics Series. Prentice Hall/Pearson Education, 2011

    Google Scholar 

  19. Thompson W. Theory of Vibration with Applications. Technology & Engineering. Fourth Edition. Taylor & Francis, 1996

    Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, The University of Pardisan Mazandaran, Babol, Iran

    M. R. Akbari

  2. Department of Chemical Engineering, University of Tehran, P.O. Box 66456-43516, Tehran, Iran

    M. R. Akbari

  3. Department of Mechanical Engineering, Babol University of Technology, P.O. Box 484, Babol, Iran

    D. D. Ganji & Sayyid H. Hashemi Kachapi

  4. Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, P.O. Box 48161-19318, Iran

    A. R. Ahmadi

Authors
  1. M. R. Akbari
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  2. D. D. Ganji
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  3. A. R. Ahmadi
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  4. Sayyid H. Hashemi Kachapi
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Correspondence to D. D. Ganji.

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Akbari, M.R., 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. Front. Mech. Eng. 9, 58–70 (2014). https://doi.org/10.1007/s11465-014-0289-7

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  • DOI: https://doi.org/10.1007/s11465-014-0289-7

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