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A New Method for the Frequency Response Curve and Its Unstable Region of a Strongly Nonlinear Oscillator

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Abstract

In order to determine the frequency response curve and its unstable region of a strongly nonlinear oscillator, a new method is proposed. This method is based on splitting the system parameters and introducing some unknown parameters into the system. The evaluation of the introduced parameters is done by optimizing the cumulative equation error induced by multiple-scales solution. The Duffing oscillator, the Helmholtz–Duffing oscillator, and an oscillator with both nonlinear restoring and nonlinear inertial forces are analyzed as examples to reveal the validity of the proposed method. The frequency response curves obtained by numerical continuation method are adopted to compare with those obtained by the proposed method and the conventional multiple-scales method. The unstable regions obtained by the harmonic balance method are adopted to examine those obtained by the conventional multiple-scales method and the proposed method. The efficiency of the proposed method is tested by comparing the computational time of each method.

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References

  1. Baily, E.: Steady-state harmonic analysis of nonlinear networks, Ph.D. thesis, Stanford University (1968). https://books.google.com/books?id=lNZYnQEACAAJ

  2. Lindenlaub, J.C.: An approach for finding the sinusoidal steady state response of nonlinear systems. In: Proceedings of the 7th Annual Allerton Conference on Circuit and System Theory, University of Illinois (1969)

    Google Scholar 

  3. Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University (1992)

    Google Scholar 

  4. Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14(4), 983–997 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  5. Odibat, Z.M.: A study on the convergence of homotopy analysis method. Appl. Math. Comput. 217, 782–789 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2003–2016 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  7. Burton, T.D., Rahman, Z.: On the multiple-scale analysis of strongly non-linear forced oscillators. Int. J. Non-Lin. Mech. 21(2), 135–146 (1986)

    Article  Google Scholar 

  8. Chen, S., Shen, J., Sze, K.: A new perturbation procedure for limit cycle analysis in three-dimensional nonlinear autonomous dynamical systems. Nonlinear Dyn. 56(3), 255–268 (2009)

    Article  MathSciNet  Google Scholar 

  9. Cheung, Y., Chen, S., Lau, S.: A modified Lindstedt–Poincaré method for certain strongly non-linear oscillators. Int. J. Non-Lin. Mech. 26(3–4), 367–378 (1991)

    Article  Google Scholar 

  10. Wu, B., Zhou, Y., Lim, C. W., Sun, W.: Analytical approximations to resonance response of harmonically forced strongly odd nonlinear oscillators. Arch. Appl. Mech. 88(12), 2123–2134 (2018)

    Article  ADS  Google Scholar 

  11. Hayashi, C.: Nonlinear Oscillations in Physical Systems. Princeton University Press, New York (2014)

    MATH  Google Scholar 

  12. Benedettini, F., Rega, G.: Non-linear dynamics of an elastic cable under planar excitation. Int. J. Non-Lin. Mech. 22(6), 497–509 (1987)

    Article  Google Scholar 

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Acknowledgements

The results presented in this paper were obtained under the supports of the Science and Technology Development Fund of Macau (Grant No. 042/2017/A1) and the Research Committee of University of Macau (Grant No. MYRG2018-00116-FST).

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Macau, Macau SAR, China

    Hai-En Du, Guo-Kang Er & Vai Pan Iu

Authors
  1. Hai-En Du
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  2. Guo-Kang Er
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  3. Vai Pan Iu
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Corresponding author

Correspondence to Hai-En Du .

Editor information

Editors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy

    Walter Lacarbonara

  2. Department of Mechanical Engineering, University of Maryland, College Park, MD, USA

    Balakumar Balachandran

  3. Department of Physics, Lanzhou University of Technology, Lanzhou, Gansu, China

    Jun Ma

  4. Department of Electrical Engineering, Polytechnic of Porto - School of Engineering (ISEP), Porto, Portugal

    J. A. Tenreiro Machado

  5. Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary

    Gabor Stepan

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Du, HE., Er, GK., Iu, V.P. (2020). A New Method for the Frequency Response Curve and Its Unstable Region of a Strongly Nonlinear Oscillator. In: Lacarbonara, W., Balachandran, B., Ma, J., Tenreiro Machado, J., Stepan, G. (eds) Nonlinear Dynamics of Structures, Systems and Devices. Springer, Cham. https://doi.org/10.1007/978-3-030-34713-0_7

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