Research Papers

Characterizing Dynamic Transitions Associated With Snap-Through: A Discrete System

[+] Author and Article Information
R. Wiebe

Graduate Research Assistant
Department of Civil Engineering,
Duke University,
Durham, NC 27708
e-mail: rw75@duke.edu

L. N. Virgin

Department of Mechanical Engineering,
Duke University,
Durham, NC 27708
e-mail: l.virgin@duke.edu

I. Stanciulescu

Department of Civil Engineering,
Rice University,
Houston, TX 77251
e-mail: ilinca@rice.edu

S. M. Spottswood

Aerospace Engineer
e-mail: Stephen.Spottswood@wpafb.af.mil

T. G. Eason

Aerospace Engineer
e-mail: Thomas.Eason@wpafb.af.mil
Structural Mechanics Branch,
2790 D Street,
WPAFB, OH 45433-7402

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 21, 2011; final manuscript received February 10, 2012; published online June 14, 2012. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 8(1), 011010 (Jun 14, 2012) (11 pages) Paper No: CND-11-1223; doi: 10.1115/1.4006201 History: Received November 21, 2011; Revised February 10, 2012

Geometrically nonlinear structures often possess multiple equilibrium configurations. Under extreme conditions of excitation, it is possible for these structures to exhibit oscillations about and between these co-existing configurations. This behavior may have serious implications for fatigue in the context of aircraft surface panels. Snap-through is a name often given to sudden changes in dynamic behavior associated with mechanical instability (buckling). This is an often encountered problem in hypersonic vehicles in which severe thermal loading and acoustic excitation conspire to create an especially hostile environment for structural elements. In this paper, a simple link model is used, experimentally and numerically, to investigate the mechanisms of snap-through buckling from a phenomenological standpoint.

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Fig. 1

A typical curved panel, and two schematic scenarios in which such a system might exhibit a snap-through event in its force-displacement relationship. (a) Presnap, (b) post-snap, (c) limit point buckling, and (d) pitchfork bifurcation.

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Fig. 2

A single degree of freedom (SDOF) link model

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Fig. 3

(a) Photograph of experimental setup, (b) the low-friction pin joint, (c) the Scotch-yoke forcing mechanism

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Fig. 4

Identification of damping parameter β (Kg/s). (a) A typical nonlinear free decay. The points are experimental data, and the continuous line represents the numerical integration of Eq. (4) with β = 1.2. (b) Normalized average error versus β for the large amplitude time series (in part (a)). (c) Normalized average error versus β for a small amplitude time series.

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Fig. 5

Free response characteristics. (a) Force versus natural frequency (squared), (b) force versus deflection, and (c) natural frequency (squared) versus deflection. The points are experimental data, and the continuous lines are the theoretical results.

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Fig. 6

Experimental and simulated time series superimposed on the restoring force. Numerical, (a) Ω = 4.40 rad/s, and (b) Ω = 3.36 rad/s. Experimental, (c) Ω = 4.40 rad/s, and (d) Ω = 3.36 rad/s.

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Fig. 7

Numerical ((a)-(d)) and experimental ((e)-(h)) time series. (a) Ω = 4.9 rad/s, (b) Ω = 4.9 rad/s, (c) Ω = 7.6 rad/s, (d) Ω = 7.8 rad/s, (e) Ω = 4.9 rad/s, (f) Ω = 4.9 rad/s, (g) Ω = 7.9 rad/s, (h) Ω = 8.1 rad/s.

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Fig. 8

Experimental and simulated DFT’s for Figs. 7(c), 7(d), 7(g), and 7(h), respectively

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Fig. 9

Occurrence of snap-through. (a) Simulation, (b) experiment, and (c) relative dominance of co-existing attractors (simulation only). Green - nonsnap, red - P1 snap-through, and blue - higher periodic or chaotic (less frequent) snap-through. The vertical dashed red lines in (a) and (b) indicate the specific frequencies relating to Fig. 7.

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Fig. 10

Average kinetic energy as a function of forcing frequency: (a) simulation and (b) experiment

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Fig. 11

Distinction between chaotic and nonchaotic behavior based on (a) the largest Lyapunov exponent, (b) the peak- count criterion, (c) and (d) typical chaotic time series, and (e) relative dominance of chaotic behavior

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Fig. 12

Experimental LE and peak count: (a) and (b) typical linear fits for the local rate of divergence, (c) largest LE as a function of the forcing frequency, and (d) corresponding peak count result



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