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Research Papers

Switching Mechanism and Complex Motions in an Extended Fermi-Acceleration Oscillator

[+] Author and Article Information
Albert C. J. Luo

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

Yu Guo

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

J. Comput. Nonlinear Dynam 5(4), 041007 (Jul 28, 2010) (14 pages) doi:10.1115/1.4001905 History: Received December 01, 2008; Revised April 22, 2009; Published July 28, 2010; Online July 28, 2010

In this paper, an extended model of the Fermi-acceleration oscillator is presented to describe impacting chatters, grazing, and sticking between the particle (or bouncing ball) and piston. The sticking phenomenon in such a system is investigated for the first time. Even in the traditional Fermi-oscillator, such a sticking phenomenon still exists but one often ignored it. In this paper, the analytical conditions for the grazing and sticking phenomena between the particle and piston in the Fermi-acceleration oscillator are developed from the theory of discontinuous dynamical systems. Compared with existing studies, the four exact mappings are used to analyze the motion behaviors of the Fermi-oscillator instead of one or two mappings. Mapping structures formed by generic mappings are adopted for the analytical predictions of periodic motions in the Fermi-acceleration oscillator. Periodic and chaotic motions in such an oscillator are illustrated to show motion complexity and grazing and sticking mechanism. Once the masses of the ball and primary mass are in the same quantity level, the model presented in this paper will be very useful and significant. This idea can apply to a system possessing two independent oscillators with impact, such as gear transmission systems, bearing systems, and time-varying billiard systems.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Mechanical model

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Figure 2

Domains and boundaries without stick in the absolute frame: (a) particle and (b) piston

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Figure 7

Numerical prediction of bifurcation scenario for switching points versus the mass ratio (m1/m2): (a) switching displacement and (b) switching velocity of the particle; (c) switching displacement and (d) switching velocity of the piston; and (e) switching phase of the system. (k/m2=10,c/m2=6,e2=0.9,e2=0.7,h=1,Q/m2=0.5,Ω=22).

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Figure 8

Analytical prediction of periodic motion for the excitation frequency of m1/m2∊(0,0.6): (a) switching displacement and (b) switching velocity of the particle; (c) switching displacement and (d) switching velocity of the piston; and (e) switching phase of the system. (k/m2=10,c/m2=6,e1=0.9,e2=0.7,h=1,Q/m2=0.5,Ω=22).

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Figure 9

A periodic motion (m1=0.01 and m2=1.0) with mapping structure P(21)2: (a) displacement time-history, (b) velocity time-history, and (c) phase plane of particle and (d) phase plane of piston. Solid and dashed curves indicate the motion of particle and piston, respectively. (Q=0.5,Ω=22 k=10,c=6,e1=0.9,e2=0.7,h=1,t0=0.2376356,x0(1)=x0(2)=−0.1598333,ẋ0(1)=−8.5090867,ẋ0(2)=1.9931273).

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Figure 10

Chaotic motion (m1=0.2 and m2=1.0): (a) displacement time-history and (b) velocity time-history. Poincare mapping sections: (e) switching displacement and (f) velocity versus switching phases for the particle; (g) switching displacement and (h) velocity versus switching phases for the piston. Solid and dashed curves indicate the motion of particle and piston, respectively. (Q=0.5,Ω=22 k=10,c=6,e1=0.9,e2=0.7,h=1,t0=0.0582876,x0(1)=x0(2)=−0.2250554,ẋ0(1)=−0.5202652 ẋ0(2)=−1.7001242).

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Figure 11

Chaotic motion of P3k(k→∞) (m1=0.5 and m2=1.0): (a) displacement time-history and (b) velocity time-history. Poincare mapping sections: (e) switching displacement and (f) velocity versus switching phases for the particle; (g) switching displacement and (h) velocity versus switching phases for the piston. Solid and dashed curves indicate the motion of particle and piston, respectively. (Q=0.5,Ω=22,  k=10,c=6,  e1=0.9,e2=0.7,h=1,t0=0.0582876,x0(1)=x0(2)=−0.2250554,ẋ0(1)=−0.9978272 ẋ0(2)=−2.1776862).

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Figure 3

Domains and boundaries with stick in the absolute frame: (a) particle and (b) piston

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Figure 4

Domains and boundaries in the relative frame: (a) (z,ż)-plane for particle, (b) (ż,z̈)-plane for particle, (c) (z,ż)-plane for piston, and (d) in (ż,z̈)-plane for piston

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Figure 5

Switching sets and generic mappings for nonstick motion in absolute coordinates: (a) particle and (b) piston

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Figure 6

Switching sets and generic mappings for stick motion in absolute coordinates: (a) particle and (b) piston

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