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Research Papers

Nonlinear Dynamic Analysis of an Icosahedron Frame which Exhibits Chaotic Behavior

[+] Author and Article Information
Lucas W. Just

Captain, USAF,
National Air and Space Intelligence Center,
Air Force Institute of Technology,
4180 Watson Way,
WPAFB, OH 45433

Anthony M. DeLuca

Assistant Professor,
Lieutenant Colonel, USAF,
Department of Aeronautics and Astronautics,
Air Force Institute of Technology,
2950 Hobson Way,
WPAFB, OH 45433-7765

Anthony N. Palazotto

Department of Aeronautics and Astronautics,
Air Force Institute of Technology,
2950 Hobson Way,
WPAFB, OH 45433-7765

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 13, 2015; final manuscript received July 6, 2016; published online September 1, 2016. Assoc. Editor: D. Dane Quinn.This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Comput. Nonlinear Dynam 12(1), 011006 (Sep 01, 2016) (10 pages) Paper No: CND-15-1373; doi: 10.1115/1.4034265 History: Received November 13, 2015; Revised July 06, 2016

The research question addressed is whether a lighter than air vehicle (LTAV), which uses an internal vacuum to become positively buoyant, can be designed to provide extended loiter for U.S. Air Force applications. To achieve a vacuum, internal gases are evacuated from the vessel, which creates a dynamic response in the supporting structural frame. This paper considers the frame of an icosahedron shaped LTAV subject to external atmospheric pressure evacuated at varying rates. A static finite element analysis documented in previous research revealed a snapback phenomenon in the frame members under certain loading conditions. A nonlinear chaotic response was observed when a dynamic analysis was conducted with the same boundary conditions used in the static analysis. The chaotic response for a variety of boundary conditions, generated by varying the rate of evacuation, similar to a ramp input, is determined. An analysis of the dynamic response is determined nonlinearly using a method that relies on a reference point distribution of external pressures to distribute the surface force across the frame. A novel method of combining the power spectral density with a Lyapunov exponent was used to determine the degree of nonlinearity and chaotic response for each boundary condition examined.

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Figures

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Fig. 1

Icosahedron frame with boundary conditions (top and bottom vertex constrained in the x and y direction) and external applied loading

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Fig. 2

Boundary condition 2 (antisymmetric condition) load–displacement curve exhibiting snapback behavior

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Fig. 3

Boundary condition 3 (symmetric condition) load–displacement curve [2]

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Fig. 4

Ramp input and response for a 1D spring-mass system

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Fig. 5

Single pendulum system (top) and phase-plane trajectory (bottom) [7]

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Fig. 6

Double pendulum system with different initial conditions (left) and the trajectories of the two points corresponding to each pendulum (right) [8]

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Fig. 7

Phase space plot of single pendulum motion decaying to attractor [6]

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Fig. 8

Load one displacement versus time response, PSD plot, phase-plane trajectory, and Lyapunov exponent convergence (clockwise from top-left) for nonchaotic, purely periodic motion

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Fig. 9

Load number 1 delay reconstructed attractor

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Fig. 10

Load four displacement versus time response, PSD plot, phase-plane trajectory, and Lyapunov exponent convergence (clockwise from top-left) for slightly chaotic, perturbed periodic motion

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Fig. 11

Load number four delay reconstructed attractor

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Fig. 12

Load three displacement versus time response, PSD plot, phase-plane trajectory, and Lyapunov exponent convergence (clockwise from top-left) for highly chaotic motion

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Fig. 13

Load number 3 delay reconstructed attractor

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