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TECHNICAL PAPERS

On the Mechanism of Stick and Nonstick, Periodic Motions in a Periodically Forced, Linear Oscillator With Dry Friction

[+] Author and Article Information
Albert C. Luo1

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

Brandon C. Gegg

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

1

To whom all correspondence should be addressed.

J. Vib. Acoust 128(1), 97-105 (Apr 28, 2005) (9 pages) doi:10.1115/1.2128644 History: Received January 22, 2005; Revised April 28, 2005

In this paper, the dynamics mechanism of stick and nonstick motion for a dry-friction oscillator is discussed. From the theory of Luo in 2005 [Commun. Nonlinear Sci. Numer. Simul., 10, pp. 1–55], the conditions for stick and nonstick motions are achieved. The stick and nonstick periodic motions are predicted analytically through the appropriate mapping structures. The local stability and bifurcation conditions for such periodic motions are obtained. The stick motions are illustrated through the displacement, velocity, and force responses. This investigation provides a better understanding of stick and nonstick motions of the linear oscillator with dry friction. The methodology presented in this paper is applicable to oscillators with nonlinear friction forces.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Mechanical model: (a) schematic model and (b) friction forces

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Figure 2

Domain partition in phase plane for oscillators with dry friction

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Figure 6

Numerical predicted bifurcation scenario: (a) switching phase and (b) switching displacement varying with excitation frequency (V=1, A0=90, d1=1, d2=0, b1=−b2=3, c1=c2=30)

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Figure 11

Displacement, velocity, and force responses of stick periodic motion for V=0 with mapping P2∘P0∘P1∘P0 for Ω=0.75 and the initial condition (Ωti,xi,ẋi)≈(3.4875,−2.9247,0): (a) phase plane, (b) force distribution along displacement, (c) force distribution along velocity, (d) displacement-time history, (e) velocity-time history and (f) forces-time history (A0=90, d1=1, d2=0, b1=−b2=3, c1=c2=30).

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Figure 3

Vector fields of stick motion with (a) vanishing only and (b) onset and vanishing for belt speed V>0. The gray and dark-filled circular symbols represent the vanishing and onset of stick motion, respectively.

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Figure 4

(a) Regular and (b) stick mappings for oscillators with dry friction

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Figure 5

Periodic motion with stick from ∂Ω21 to Ω1

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Figure 7

Analytical prediction of (a) switching phase and (b) switching displacement for mapping P2∘P1 (V=1, A0=90, d1=1, d2=0, b1=−b2=3, c1=c2=30)

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Figure 8

Eigenvalue analysis: (a) magnitude and (b) real part of two eigenvalues for mapping P2∘P1 (V=1, A0=90, d1=1, d2=0, b1=−b2=3, c1=c2=30)

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Figure 9

Periodic response of mapping P2∘P1: (a) phase plane and (b) force distribution for Ω=2.25 and the initial conditions (Ωti,xi,ẋi)≈(3.1143,−3.3583,1) (V=1, A0=90, d1=1, d2=0, b1=−b2=3, c1=c2=30)

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Figure 10

Displacement, velocity, and force responses of stick periodic motion for V=1 with mapping P2∘P0∘P1∘P0 for Ω=0.45 and the initial condition (Ωti,xi,ẋi)≈(3.6932,−2.5506,1): (a) phase plane, (b) force distribution along displacement, (c) force distribution along velocity, (d) displacement-time history, (e) velocity-time history and (f) forces-time history (A0=90, d1=1, d2=0, b1=−b2=3, c1=c2=30).

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