ANALYSIS OF A DAMPED FRACTAL SYSTEM USING THE ANCIENT CHINESE ALGORITHM AND THE TWO-SCALE FRACTAL DIMENSION TRANSFORM

ELíAS-ZÚÑIGA Alex, Oscar Martã­nez-Romero (), Daniel Olvera Trejo () and Luis Manuel Palacios-Pineda
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ELíAS-ZÚÑIGA Alex: Mechanical Engineering and Advanced Materials Department and Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico
Oscar Martã­nez-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

FRACTALS (fractals), 2022, vol. 30, issue 09, 1-8

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.

Keywords: Two-Scale Fractal Dimension Transform; Hes Frequency Amplitude Formulation; Ancient Chinese Algorithm; Damped Fractal Oscillators; Fractal Plasma Physics Equation (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22501730

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