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TECHNICAL PAPERS

Grazing and Chaos in a Periodically Forced, Piecewise Linear System

[+] Author and Article Information
Albert C. Luo

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805aluo@siue.edu

J. Vib. Acoust 128(1), 28-34 (Jul 01, 2005) (7 pages) doi:10.1115/1.2149390 History: Received March 17, 2005; Revised July 01, 2005

The criteria for the grazing bifurcation of a periodically forced, piecewise linear system are developed and the initial grazing manifolds are obtained. The initial grazing manifold is invariant. The grazing flows are illustrated to verify the analytic prediction of grazing. The mechanism of the strange attractors fragmentation caused by the grazing is discussed, and an illustration of the fragmentized strange attractor is given through the Poincaré mapping. This fragmentation phenomenon exists extensively in nonsmooth dynamical systems. The mathematical structure of the fragmentized strange attractors should be further developed.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Subdomains, boundaries, and phase portraits near the static equilibrium points (±E,0) on the boundaries

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Figure 2

Switching sections and basic mappings in phase plane

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Figure 3

The grazing bifurcation conditions for mappings P1(yi⩾0) and P3(yi⩽0) represented by the solid and dashed curves, respectively (n=1, a=20, c=100, E=1, d=0.5)

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Figure 4

The grazing bifurcation conditions for (a) mappings P2(yi⩽0) and P4(yi⩾0) and (b) mappings P5(yi⩽0) and P6(yi⩾0). The mapping {P2,P5} and {P3,P6} are represented by the solid and dashed curves, respectively (n=1, a=20, c=100, E=1, d=0.5).

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Figure 5

The grazing flows in domains Ω1 and Ω3 for (a) mapping P1 (xi=1, (yi,Ωti)≈(1,3.592), (2,3.256), and (3,3.125)); (b) mapping P3 (xi=−1, (yi,Ωti)≈(−1,0.449), (−2,0.114) and (−3,6.267)). a=20, c=100, E=1, d=0.5, Ω=10.

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Figure 6

The grazing flows in the domain Ω2: (a) mapping P2 (xi=1, (yi,Ωti)≈(0,2.71), (−2,3.20), (−4,3.63), (−6,4.07)); (b) mapping P4 (xi=−1, (yi,Ωti)≈(0,5.85), (2,0.06), (−4,0.47), (−6,0.93)). a=20, c=100, E=1, d=0.5, Ω=10.

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Figure 7

The local mapping structures of mapping structures P4(36)m232(15)m11 in the switching sections. The gray-filled areas are the switching sets for the corresponding mappings. The local and global mappings are represented by the solid and dashed arrows, respectively.

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Figure 8

The initial grazing switching manifolds for the six single mappings in the switching sections. The gray-filled areas are the switching sets of P4(36)m232(15)m11 for the corresponding mappings. The dashed lines are the initial grazing, switching manifolds for given parameters.

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Figure 9

The grazing in chaotic motion pertaining to mappings P4(36)m232(15)m11 in the switching sections: (a) local mapping grazing and (b) global mapping grazing. The red and hollow symbols represent the initial and ending points respectively, for grazing. The solid and dashed arrows depict the mapping routines before and after grazing, respectively.

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Figure 10

Chaotic motion associated with mappings (P4(36)321, P4321, and P432(15)1): (a) subsets of switching plane Σ+ (Σ++ and Σ−+) and (b) subsets of switching plane Σ− (Σ+− and Σ−−). a=20, c=100, E=1, d=0.5, Ωti≈6.1117, xi=1, yi≈6.5251, Ω=2.1. The circular symbols are relative to grazing.

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