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RESEARCH PAPERS

Co-dimension-Two Grazing Bifurcations in Single-Degree-of-Freedom Impact Oscillators

[+] Author and Article Information
Phanikrishna Thota

Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061thota@vt.edu

Xiaopeng Zhao

Department of Biomedical Engineering, Duke University, Durham, NC 27708xzhao@duke.edu

Harry Dankowicz

Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801danko@uiuc.edu

J. Comput. Nonlinear Dynam 1(4), 328-335 (Apr 18, 2006) (8 pages) doi:10.1115/1.2338658 History: Received January 08, 2006; Revised April 18, 2006

Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and nonimpacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuity-mapping technique are used to explore the possible influence on the global dynamics of the smooth co-dimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear micro-electro-mechanical systems (MEMS) oscillator.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Grazing curve for the linear impact oscillator in (ω,α) space. Here, ζ=0.05 and δ=1.

Grahic Jump Location
Figure 2

Grazing curve for the nonlinear MEMS oscillator in (ω,V) space. Here, ζ=0.02 and δ=0.5. Here, and in later figures, solid lines refer to stable orbits and dashed lines refer to unstable orbits.

Grahic Jump Location
Figure 3

Co-dimension-two points ξn=0 for ω⩾ω1*

Grahic Jump Location
Figure 4

A continuous grazing bifurcation is achieved in the linear oscillator with ω=21−ζ2, ζ=0.05, and δ=1, for which ξ1=0. Here, and in later figures, the penetration of a point on P is defined as −hD and G denotes the grazing bifurcation.

Grahic Jump Location
Figure 5

Sketch of the bifurcation curves associated with impacting trajectories emanating from the co-dimension-two bifurcation point at ω≈0.0954 (scenario 1), ω≈0.1009 (scenario 2), and ω≈0.1392 (scenario 3) denoted by COD-2. Here, and in the following figures, PD denotes period-doubling bifurcations, SN denotes saddle-node bifurcations, CR denotes crisis bifurcations (e.g., homoclinic bifurcations) involving an impacting chaotic attractor. The (ω,V)-parameter space is divided into five regions according to the type of solutions: A, no stable or unstable period-1 orbits; B, coexistence of an unstable period-1 impacting orbit and an unstable nonimpacting period-1 orbit; C, coexistence of two unstable impacting period-1 orbits; D, coexistence of an impacting chaotic attractor or periodic attractor of period >1 and unstable impacting period-1 orbit(s); E, coexistence of a stable impacting period-1 orbit and an unstable impacting periodic orbit; and F, coexistence of a stable impacting period-1 orbit and an unstable nonimpacting periodic-1 orbit.

Grahic Jump Location
Figure 6

Bifurcation diagrams showing impacting and nonimpacting solutions corresponding to sweeps a (top panel, Δω=−5×10−5) and b (middle and bottom panels, Δω=3×10−6) for scenario 1. Bottom panel is the blow-up of the middle panel near the SN point.

Grahic Jump Location
Figure 7

Bifurcation diagrams showing impacting and nonimpacting solutions corresponding to sweeps a (top panel, Δω=−7×10−5) and b (bottom panel, Δω=7×10−5) for scenario 2

Grahic Jump Location
Figure 8

Bifurcation diagrams showing impacting and nonimpacting solutions corresponding to sweeps a (top panel, Δω=−5×10−5) and b (middle and bottom panels, Δω=10−5) for scenario 3. Bottom panel is the blowup of the middle panel near the grazing point G.

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