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TECHNICAL PAPERS

# Arbitrary Periodic Motions and Grazing Switching of a Forced Piecewise Linear, Impacting Oscillator

[+] Author and Article Information
Albert C. Luo1

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

Lidi Chen

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

1

Corresponding author.

J. Vib. Acoust 129(3), 276-284 (Nov 01, 2006) (9 pages) doi:10.1115/1.2424971 History: Received June 27, 2005; Revised November 01, 2006

## Abstract

The grazing bifurcation and periodic motion switching of the harmonically forced, piecewise linear system with impacting are investigated. The generic mappings relative to the discontinuous boundaries of this piecewise system are introduced. Based on such mappings, the corresponding grazing conditions are obtained. The mapping structures are developed for the analytical prediction of periodic motions in such a system. The local stability and bifurcation conditions for specified periodic motions are obtained. The regular and grazing, periodic motions are illustrated. The grazing is the origin of the periodic motion switching for this system. Such a grazing bifurcation cannot be estimated through the local stability analysis. This model is applicable to prediction of periodic motions in nonlinear dynamics of gear transmission systems.

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## Figures

Figure 1

Mechanical model of the piecewise linear impacting oscillator

Figure 2

Phase domain partition and boundaries

Figure 3

Switching sections and basic mappings in phase plane

Figure 4

A mapping structure for a periodic motion

Figure 5

Analytical prediction of periodic motion with P6431: (a) switching phase; and (b) switching velocity varying with excitation frequency (m1,3=1, m2=5, E1,3=1, Q0=500, k1,3=2000, k2=100, d1,2,3=0.2)

Figure 6

Analytical prediction of periodic motion with P6(45)43(12)1: (a) switching phase; and (b) switching velocity varying with excitation frequency (m1,3=1, m2=5, E1,3=1, Q0=500, k1,3=2000, k2=100, d1,2,3=0.2)

Figure 7

Analytical prediction of periodic motion with P6(45)243(12)21: (a) switching phase; and (b) switching velocity varying with excitation frequency (m1,3=1, m2=5, E1,3=1, Q0=500, k1,3=2000, k2=100, d1,2,3=0.2)

Figure 8

Phase trajectories and displacement responses of the stable, symmetric, periodic motions: (a),(b) P6(45)43(12)1 with initial conditions (Ω=3.6, Ωti≈6.0850, xi=1, yi≈−6.2583); and (c),(d) P6(45)243(12)21 with initial conditions (Ω=2.35, Ωti≈5.2385, xi=1, yi≈8.2854); (m1,3=1, m2=5, E1,3=1, Q0=500, k1,3=2000, k2=100, d1,2,3=0.2)

Figure 9

Phase trajectories and displacement responses for the two stable, asymmetric, periodic motions pertaining to P6(45)43(12)1 (Ω=3.35 and xi=1): (a),(b) lower asymmetry (Ωti≈5.2270, yi≈9.6488); and (c),(d) upper asymmetry (Ωti≈5.5227, yi≈10.3250); (m1,3=1, m2=5, E1,3=1, Q0=500, k1,3=2000, k2=100, d1,2,3=0.2)

Figure 10

Phase trajectories and displacement responses for the two stable, asymmetric, periodic motions pertaining to P6(45)243(12)21 (Ω=2.3 and xi=1): (a),(b) lower asymmetry (Ωti≈5.1670, yi≈8.3312); and (c),(d) upper asymmetry (Ωti≈5.3255, yi≈9.0293); (m1,3=1, m2=5, E1,3=1, Q0=500, k1,3=2000, k2=100, d1,2,3=0.2)

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