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RESEARCH PAPERS

# Periodic Motions in a Periodically Forced Oscillator Moving on an Oscillating Belt With Dry Friction

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

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

J. Comput. Nonlinear Dynam 1(3), 212-220 (Mar 22, 2006) (9 pages) doi:10.1115/1.2198874 History: Received June 27, 2005; Revised March 22, 2006

## Abstract

In this paper, periodic motion in an oscillator moving on a periodically oscillating belt with dry friction is investigated. The conditions of stick and nonstick motions for such an oscillator are obtained in the relative motion frame, and the grazing and stick (or sliding) bifurcations are presented as well. The periodic motions are predicted analytically and numerically, and the analytical prediction is based on the appropriate mapping structures. The eigenvalue analysis of such periodic motions is carried out. The periodic motions are illustrated through the displacement, velocity, and force responses in the absolute and relative frames. This investigation provides an efficient method to predict periodic motions of such an oscillator involving dry friction. The significance of this investigation lies in controlling motion of such a friction-induced oscillator in industry.

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Copyright © 2006 by American Society of Mechanical Engineers
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## Figures

Figure 9

Responses of nonstick periodic motion relative to mapping P021: (a) absolute phase trajectory and (b) relative force. (A0=80, Ω=1, V0=0.25, V1=1, d1=1, d2=0, b1=−b2=0.5, c1=c2=30). The initial conditions are (xi,ẋi,Ωti)≈(−2.0348,1.2478,3.8999). The solid and dotted curves represent the oscillator’s responses and belt responses, respectively. The red symbols denote switching points for the passable motion.

Figure 8

Responses of nonstick periodic motion relative to mapping P21: (a) phase trajectory and (b) relative force (A0=99, Ω=1, V0=0.25, V1=1, d1=1, d2=0, b1=−b2=0.5, c1=c2=30). The initial conditions are (xi,ẋi,Ωti)≈(−3.3510,1.2311,3.3750). The solid and dotted curves represent the oscillator’s responses and belt responses, respectively. The red symbols denote switching points for the passable motion.

Figure 7

Eigenvalue analysis of periodic motions varying with excitation amplitude: (a) magnitude and (b) real part (Ω=1, V0=0.25, d1=1, d2=0, d2=0, b1=−b2=0.5, c1=c2=30). The dashed and dashed-dotted vertical lines are stick and grazing bifurcations, respectively.

Figure 6

Analytical and numerical predictions of switching sets varying with excitation amplitude: (a) switching displacement and (b) switching phase (Ω=1, V0=0.25, V1=1, d1=1, d2=0, b1=−b2=0.5, c1=c2=30). The dashed and dashed-dotted vertical lines are stick and grazing bifurcations, respectively. The diamond symbols and solid curves represent the analytical and numerical predictions, respectively.

Figure 5

Basic mappings in (a) absolute and (b) relative frames for a linear oscillator with dry-friction

Figure 4

Particles switching on the oscillating belt surface in absolute phase plane. The particles (p1,p2,p3) on the belts are represented by white, yellow, and green circular symbols, respectively. The red circular symbol is the oscillator location.

Figure 3

Phase plane partition in (a) absolute and (b) relative phase planes

Figure 2

(a) Friction forces and (b) the transport speed of the oscillating belt

Figure 1

Mechanical model of the oscillator with dry friction

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