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A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

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  • Chein-Shan Liu

    (Center of Excellence for Ocean Engineering, Center of Excellence for the Oceans, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Yung-Wei Chen

    (Department of Marine Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

Abstract

In order to improve the Lindstedt-Poincaré method to raise the accuracy and the performance for the application to strongly nonlinear oscillators, a new analytic method by engaging in advance a linearization technique in the nonlinear differential equation is developed, which is realized in terms of a weight factor to decompose the nonlinear term into two sides. We expand the constant preceding the displacement in powers of the introduced parameter so that the coefficients can be determined to avoid the appearance of secular solutions. The present linearized Lindstedt-Poincaré method is easily implemented to provide accurate higher order analytic solutions of nonlinear oscillators, such as Duffing and van Der Pol nonlinear oscillators. The accuracy of analytic solutions is evaluated by comparing to the numerical results obtained from the fourth-order Runge-Kotta method. The major novelty is that we can simplify the Lindstedt-Poincaré method to solve strongly a nonlinear oscillator with a large vibration amplitude.

Suggested Citation

  • Chein-Shan Liu & Yung-Wei Chen, 2021. "A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations," Mathematics, MDPI, vol. 9(23), pages 1-17, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:23:p:3070-:d:690806
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    References listed on IDEAS

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    1. Chein-Shan Liu, 2021. "Linearized Homotopy Perturbation Method for Two Nonlinear Problems of Duffing Equations," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 13(6), pages 1-10, December.
    2. Gadella, M. & Giacomini, H. & Lara, L.P., 2015. "Periodic analytic approximate solutions for the Mathieu equation," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 436-445.
    3. He, Ji-Huan & Abdou, M.A., 2007. "New periodic solutions for nonlinear evolution equations using Exp-function method," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1421-1429.
    4. Wei, Zhouchao & Zhu, Bin & Yang, Jing & Perc, Matjaž & Slavinec, Mitja, 2019. "Bifurcation analysis of two disc dynamos with viscous friction and multiple time delays," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 265-281.
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