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Analysis Of A Damped Fractal System Using The Ancient Chinese Algorithm And The Two-Scale Fractal Dimension Transform

Author

Listed:
  • ALEX ELíAS-ZÚÑIGA

    (Mechanical Engineering and Advanced Materials Department and Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico)

  • OSCAR MARTíNEZ-ROMERO

    (Mechanical Engineering and Advanced Materials Department and Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico)

  • DANIEL OLVERA TREJO

    (Mechanical Engineering and Advanced Materials Department and Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico)

  • LUIS MANUEL PALACIOS-PINEDA

    (Mechanical Engineering and Advanced Materials Department and Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico†Tecnológico Nacional de México/Instituto, Tecnológico de Pachuca, Carr. México-Pachuca Km 87.5, Pachuca, Hidalgo, Código Postal 42080, Mexico)

Abstract

This paper investigates the applicability of the ancient Chinese algorithm jointly with the two-scale fractal dimension transform to find the frequency–amplitude relationship of fractal equations of motion with and without damping terms. Analytical results show that for a fractal equation of motion without damping terms, the oscillation amplitudes do not exhibit decaying effects. However, when damping terms are included, the fractal parameter tends to shift the decaying oscillation amplitudes that decrease faster with time for fractal values less than one. This paper provides an efficient tool for finding the amplitude–frequency relationship of damped fractal oscillators. To illustrate the solution process, the steady-state solution of the fractal equation of motion that arises in plasma physics is derived. The proposed approach elucidates the applicability of He’s formulation jointly with the two-scale fractal calculus to find the frequency–amplitude of fractal systems with and without damping terms.

Suggested Citation

  • Alex Elã­As-Zãšã‘Iga & Oscar Martã­Nez-Romero & Daniel Olvera Trejo & Luis Manuel Palacios-Pineda, 2022. "Analysis Of A Damped Fractal System Using The Ancient Chinese Algorithm And The Two-Scale Fractal Dimension Transform," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(09), pages 1-8, December.
    • Handle: RePEc:wsi:fracta:v:30:y:2022:i:09:n:s0218348x22501730
    DOI: 10.1142/S0218348X22501730
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