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Fractal equilibrium configuration of a mechanically loaded binary tree

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  • Rodríguez-Cuadrado, Javier
  • San Martín, Jesús

Abstract

In this paper we study the equilibrium mechanics problem that originates in a binary tree with infinite levels subjected to loads on its topmost branches. The application of the laws of mechanics to find the equilibrium configuration shows that the functional forms of the vertical and horizontal displacements of its end nodes converge to a Takagi curve and a linear combination of inverses of β-Cantor functions respectively as the number of levels tend to infinity. As a consequence, the shape of the canopy results from the combination of these two emerging fractals that were not present in the unloaded tree. Besides, our study also shows that the analytical expressions of the emerging fractals depend on the mechanical properties of the binary tree, indicating that the binary tree is a link between these two emerging fractals. In addition, we prove that the fractal dimensions of Takagi and β-Cantor are related in this model.

Suggested Citation

  • Rodríguez-Cuadrado, Javier & San Martín, Jesús, 2021. "Fractal equilibrium configuration of a mechanically loaded binary tree," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    • Handle: RePEc:eee:chsofr:v:152:y:2021:i:c:s0960077921007694
    DOI: 10.1016/j.chaos.2021.111415
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    References listed on IDEAS

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    1. Ionescu, Clara & Kelly, James F., 2017. "Fractional calculus for respiratory mechanics: Power law impedance, viscoelasticity, and tissue heterogeneity," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 433-440.
    2. Carpinteri, Alberto & Pugno, Nicola & Sapora, Alberto, 2009. "Asymptotic analysis of a von Koch beam," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 795-802.
    3. Kalmár-Nagy, Tamás & Varga, Árpád, 2019. "Complexity analysis of turbulent flow around a street canyon," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 102-117.
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