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Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method

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Abstract

A homotopy analysis method (HAM) is presented for the primary resonance of multiple degree-of-freedom systems with strong non-linearity excited by harmonic forces. The validity of the HAM is independent of the existence of small parameters in the considered equation. The HAM provides a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter. Two examples are presented to show that the HAM solutions agree well with the results of the modified Linstedt-Poincaré method and the incremental harmonic balance method.

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References

  1. Melvin, P. J. On the construction of Poincaré-Lindstedt solutions: the nonlinear oscillator equation. SIAM Journal on Applied Mathematics 33(1), 161–194 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  2. Nayfeh, A. H. Perturbation Methods, John Wiley and Sons, New York (1973)

    MATH  Google Scholar 

  3. Larionov, G. S. and Khoang, V. T. The method of averaging in nonlinear dynamical problems occurring in the theory of viscoelasticity. Mechanics of Composite Materials 8(1), 31–35 (1972)

    Google Scholar 

  4. Kakutani, T. and Sugimoto, N. Krylov-Bogoliubov-Mitropolsky method for nonlinear wave modulation. Physics of Fluids 17(8), 1617–1625 (1974)

    Article  MathSciNet  Google Scholar 

  5. Cheung, Y. K., Chen, S. H., and Lau, S. L. Modified Linstedt-Poincaré method for certain strongly nonlinear oscillators. International Journal of Non-Linear Mechanics 26(3), 367–378 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Pierre, C. and Dowell, E. H. A study of dynamic instability of plates by an extended incremental harmonic balance method. Journal of Applied Mechanics 52(3), 693–697 (1985)

    Article  MATH  Google Scholar 

  7. Andrianov, I. V., Danishevs’kyy, V. V., and Awrejcewicz, J. An artificial small perturbation parameter and nonlinear plate vibrations. Journal of Sound and Vibration 283(3), 561–571 (2005)

    Article  Google Scholar 

  8. Karmishin, A. V., Zhukov, A. I., and Kolosov, V. G. Methods of Dynamics Calculation and Testing for Thin-Walled Structures, Mashinostroyenie, Moscow (1990)

    Google Scholar 

  9. Adomian, G. A new approach to the heat equation—an application of the decomposition method. Journal of Mathematical Analysis and Applications 113(1), 202–209 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  10. Liao, S. J. Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton (2003)

    Book  Google Scholar 

  11. Liao, S. J. A new branch of solutions of boundary-layer flows over an impermeable stretched plate. International Journal of Heat and Mass Transfer 48(12), 2529–2539 (2005)

    Article  MATH  Google Scholar 

  12. Liao, S. J. An explicit, totally analytic approximate solution for Blasius’ viscous flow problems. International Journal of Non-Linear Mechanics 34(4), 759–778 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Liao, S. J. Series solutions of unsteady boundary-layer flows over a stretching flat plate. Studies in Applied Mathematics 117(3), 239–264 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Feng, S. D. and Chen, L. Q. Homotopy analysis approach to Duffing harmonic oscillator. Applied Mathematics and Mechanics (English Edition) 30(9), 1015–1020 (2009) DOI 10.1007/s10483-009-0902-7

    Article  MathSciNet  Google Scholar 

  15. Liao, S. J. An approximate solution technique which does not depend upon small parameters: a special example. International Journal of Non-Linear Mechanics 30(3), 371–380 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Liao, S. J. An approximate solution technique which does not depend upon small parameters-II, an application in fluid mechanics. International Journal of Non-Linear Mechanics 32(5), 815–822 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Liao, S. J. Homotopy analysis method: a new analytic method for nonlinear problems. Applied Mathematics and Mechanics (English Edition) 19(10), 957–962 (1998) DOI 10.1007/BF02457955

    Article  MATH  MathSciNet  Google Scholar 

  18. Wen, J. M. and Cao, Z. C. Sub-harmonic resonances of nonlinear oscillations with parametric excitation by means of the homotopy analysis method. Physics Letters A 371(5–6), 427–431 (2007)

    Article  Google Scholar 

  19. Tan, Y. and Abbasbandy, S. Homotopy analysis method for quadratic Riccati differential equation. Communication Nonlinear Science and Numerical Simulation 13(3), 539–546 (2008)

    Article  MATH  Google Scholar 

  20. Abbasbandy, S. Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method. Chemical Engineering Journal 136(2–3), 144–150 (2008)

    Article  Google Scholar 

  21. Li, S., Wang, B., and Hu, J. Z. Homotopy solution of the inverse generalized eigenvalue problems in structural dynamics. Applied Mathematics and Mechanics (English Edition) 25(5), 580–586 (2004) DOI 10.1007/BF02437606

    Article  MATH  Google Scholar 

  22. Chen, S. H. and Cheung, Y. K. A modified Lindstedt-Poincaré method for a strongly nonlinear two degree-of-freedom system. Journal of Sound and Vibration 193(4), 751–762 (1996)

    Article  MathSciNet  Google Scholar 

  23. Cheung, Y. K., Chen, S. H., and Lau, S. L. Application of the incremental harmonic balance method to cubic non-linearity systems. Journal of Sound and Vibration 140(2), 273–286 (1990)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mechanical Engineering and Automation School, Northeastern University, Shenyang, 110004, P. R. China

    Pei-xin Yuan  (原培新)

  2. College of Science, Northeastern University, Shenyang, 110004, P. R. China

    Yong-qiang Li  (李永强)

Authors
  1. Pei-xin Yuan  (原培新)
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  2. Yong-qiang Li  (李永强)
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Correspondence to Pei-xin Yuan  (原培新).

Additional information

Communicated by Li-qun CHEN

Project supported by the Fundamental Research Funds for the Central Universities (No. N090405009)

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Yuan, Px., Li, Yq. Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method. Appl. Math. Mech.-Engl. Ed. 31, 1293–1304 (2010). https://doi.org/10.1007/s10483-010-1362-6

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  • DOI: https://doi.org/10.1007/s10483-010-1362-6

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