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Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator

Author

Listed:
  • Alvaro H. Salas

    (FIZMAKO Research Group, Department of Mathematics and Statistics, Universidad Nacional de Colombia, Manizales 170001, Colombia)

  • Ma’mon Abu Hammad

    (Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan)

  • Badriah M. Alotaibi

    (Department of Physics, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia)

  • Lamiaa S. El-Sherif

    (Department of Physics, College of Arts and Science in Wadi Al-Dawaser, Prince Sattam bin Addulaziz University, Wadi-Dawaser 11991, Saudi Arabia
    Department of Physics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt)

  • Samir A. El-Tantawy

    (Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt
    Research Center for Physics (RCP), Department of Physics, Faculty of Science and Arts, Al-Mikhwah, Al-Baha University, Al Bahah 1988, Saudi Arabia)

Abstract

In this investigation, some analytical solutions to both conserved and non-conserved rotational pendulum systems are reported. The exact solution to the conserved oscillator (unforced, undamped rotational pendulum oscillator), is derived in the form of a Jacobi elliptical function. Moreover, an approximate solution for the conserved case is obtained in the form of a trigonometric function. A comparison between both exact and approximate solutions to the conserved oscillator is examined. Moreover, the analytical approximations to the non-conserved oscillators including the unforced, damped rotational pendulum oscillator and forced, damped rotational pendulum oscillator are obtained. Furthermore, all mentioned oscillators (conserved and non-conserved oscillators) are linearized, and their exact solutions are derived. In addition, all obtained approximations are compared with the four-order Runge–Kutta (RK4) numerical approximations and with the exact solutions to the linearized oscillators. The obtained results can help several authors for discussing and interpreting their results.

Suggested Citation

  • Alvaro H. Salas & Ma’mon Abu Hammad & Badriah M. Alotaibi & Lamiaa S. El-Sherif & Samir A. El-Tantawy, 2022. "Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator," Mathematics, MDPI, vol. 10(21), pages 1-13, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4000-:d:956052
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    References listed on IDEAS

    as
    1. D. V. Hieu & N. Q. Hai & D. T. Hung, 2018. "The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators," Journal of Applied Mathematics, Hindawi, vol. 2018, pages 1-15, April.
    2. Bezziou, Mohamed & Jebril, Iqbal & Dahmani, Zoubir, 2021. "A new nonlinear duffing system with sequential fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    3. Weaam Alhejaili & Alvaro H. Salas & Samir A. El-Tantawy, 2022. "Novel Approximations to the (Un)forced Pendulum–Cart System: Ansatz and KBM Methods," Mathematics, MDPI, vol. 10(16), pages 1-12, August.
    Full references (including those not matched with items on IDEAS)

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