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Research Papers

Analytical Predication of Complex Motion of a Ball in a Periodically Shaken Horizontal Impact Pair

[+] Author and Article Information
Yu Guo

Department of Mechanical and Industrial Engineering,  Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805

Albert C. J. Luo1

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

1

Corresponding author.

J. Comput. Nonlinear Dynam 7(2), 021001 (Dec 22, 2011) (9 pages) doi:10.1115/1.4004884 History: Received March 17, 2011; Revised July 31, 2011; Published December 22, 2011; Online December 22, 2011

In this paper, complex motions of a ball in the horizontal impact pair with a periodic excitation are studied analytically using the theory of discontinuous dynamical system. Analytical conditions for motion switching caused by impacts are developed, and generic mapping structures are introduced to describe different periodic and chaotic motions. Analytical prediction of complex periodic motion of the ball in the periodically shaken impact pair is completed, and the corresponding stability and bifurcation analysis are also carried out. Numerical illustrations of periodic and chaotic motions are given.

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Copyright © 2012 by American Society of Mechanical Engineers
Topics: Motion , Bifurcation
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Figures

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Figure 1

Mechanical model

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Figure 2

Absolute domain and boundaries without stick

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Figure 3

Absolute domains and boundaries with stick

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Relative domains and boundaries

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Figure 5

Switching sets and generic mappings for nonstick motion in absolute coordinates

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Figure 6

Switching sets and generic mappings for stick motion in absolute coordinates

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Figure 7

Analytical prediction of varying the coefficient of restitution e. (a) Switching displacement of the ball, (b) switching velocity of the ball, and (c) switching phase. (Q0=0.4,M=1.0,m=0.001, and d=0.15.)

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Figure 8

Periodic motion with a mapping structure of P26451535 : (a) Displacement time history, (b) velocity time history, and (c) phase portrait of the ball with moving boundaries. (Q0=0.4,Ω=0.5,M=1.0,m=0.001,e=0.08, and d=0.15). The initial conditions are t0=8.44183604, x0=-0.488769302, and x·0=0.178074511.

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Figure 9

Chaotic motion: (a) displacement time history and (b) velocity time history. (Q0=0.4,Ω=0.5,M=1.0,m=0.001,e=0.2, and d=0.15). The initial conditions are t0=1.33468433, x0=0.61418775, and x·0=0.204449281.

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