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Approximate Solutions for Undamped Nonlinear Oscillations Using He’s Formulation

Author

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  • Stylianos Vasileios Kontomaris

    (Faculty of Engineering and Architecture, Metropolitan College, 15125 Athens, Greece
    BioNanoTec Ltd., 2404 Nicosia, Cyprus)

  • Georgios Chliveros

    (Faculty of Engineering and Architecture, Metropolitan College, 15125 Athens, Greece)

  • Anna Malamou

    (Radar Systems and Remote Sensing Lab, School of Electrical & Computer Engineering, National Technical University of Athens, 15780 Athens, Greece)

Abstract

Solving nonlinear oscillations is a challenging task due to the mathematical complexity of the related differential equations. In many cases, determining the oscillation’s period requires the solution of complicated integrals using numerical methods. To avoid the complexity, there are many empirical equations in the literature that can be used instead of rigorous mathematical analysis to provide an acceptable approximation. In this paper, a recently developed method, He’s formulation, is applied to find the period in many different cases of nonlinear oscillators. The cases are those of the Duffing equation, the Helmholtz nonlinear oscillator, the simple pendulum and the case of a vertical oscillation under the influence of a nonlinear elastic force. The results of the method are accurate; thus, He’s formulation is a strong tool for solving nonlinear oscillations.

Suggested Citation

  • Stylianos Vasileios Kontomaris & Georgios Chliveros & Anna Malamou, 2023. "Approximate Solutions for Undamped Nonlinear Oscillations Using He’s Formulation," J, MDPI, vol. 6(1), pages 1-12, February.
    • Handle: RePEc:gam:jjopen:v:6:y:2023:i:1:p:10-151:d:1084073
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    References listed on IDEAS

    as
    1. Cveticanin, L., 2003. "Analytic solution of the system of two coupled differential equations with the fifth-order non-linearity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 317(1), pages 83-94.
    2. L. Cveticanin & U. Kozmidis Luburic & Gy. Mester, 2018. "Periodic Motion in an Excited and Damped Cubic Nonlinear Oscillator," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-6, June.
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