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Technical Brief: Fractal Engineering and Biomedicine

Vibrations of Fractal Structures: On the Nonlinearities of Damping by Branching

[+] Author and Article Information
Peter Torab

Department of Mechanical Engineering,
Gannon University,
109 University Square, PMB 3824,
Erie, PA 16541-0001
e-mail: pctorab@gmail.com

Davide Piovesan

Biomedical Engineering Program,
Department of Mechanical Engineering,
Gannon University,
109 University Square, PMB 3251,
Erie, PA 16541-0001
e-mail: piovesan001@gannon.edu

1Corresponding author.

Manuscript received August 1, 2015; final manuscript received December 1, 2015; published online March 17, 2016. Assoc. Editor: Charalabos Doumanidis.

J. Nanotechnol. Eng. Med 6(3), 034502 (Mar 17, 2016) (7 pages) Paper No: NANO-15-1058; doi: 10.1115/1.4032224 History: Received August 01, 2015; Revised December 01, 2015

To study the effect of damping due to branching in trees and fractal structures, a harmonic analysis was performed on a finite element model using commercially available software. The model represented a three-dimensional (3D) fractal treelike structure, with properties based on oak wood and with several branch configurations. As branches were added to the model using a recursive algorithm, the effects of damping due to branching became apparent: the first natural frequency amplitude decreased, the first peak widened, and the natural frequency decreased, whereas higher frequency oscillations remained mostly unaltered. To explain this nonlinear effect observable in the spectra of branched structures, an analytical interpretation of the damping was proposed. The analytical model pointed out the dependency of Cartesian damping from the Coriolis forces and their derivative with respect to the angular velocity of each branch. The results provide some insight on the control of chaotic systems. Adding branches can be an effective way to dampen slender structures but is most effective for large deformation of the structure.

FIGURES IN THIS ARTICLE
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Copyright © 2016 by ASME
Topics: Damping , Fractals , Vibration
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Figures

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Fig. 1

Models with load conditions

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Fig. 2

Displacement spectra 0–1000 Hz

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Fig. 3

Refined spectra 0–10 Hz

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Fig. 4

Isometric view of the first three vibrational modes for each structure

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Fig. 5

Treelike structure generated by three-link kinematic chain

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Fig. 6

(a) Newtonian damping, (b) tuning mass, (c) additional mass, (d) hysteretic damping, and (e) increased mass on hysteretic damping

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