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Research Papers

Bifurcation and Stability of Periodic Motions in a Periodically Forced Oscillator With Multiple Discontinuities

[+] Author and Article Information
Albert C. Luo, Mehul T. Patel

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

J. Comput. Nonlinear Dynam 4(1), 011011 (Dec 12, 2008) (12 pages) doi:10.1115/1.3007902 History: Received April 06, 2007; Revised March 22, 2008; Published December 12, 2008

Abstract

The local stability and existence of periodic motions in a periodically forced oscillator with multiple discontinuities are investigated. The complexity of periodic motions and chaos in such a discontinuous system is often caused by the passability, sliding, and grazing of flows to discontinuous boundaries. Therefore, the corresponding analytical conditions for such singular phenomena to discontinuous boundary are presented from the local singularity theory of discontinuous systems. To develop the mapping structures of periodic motions, basic mappings are introduced, and the sliding motion on the discontinuous boundary is described by a sliding mapping. A generalized mapping structure is presented for all possible periodic motions, and the local stability and bifurcations of periodic motions are discussed. From mapping structures, the switching points of periodic motions on the boundaries are predicted analytically. Two periodic motions are presented for illustrations of the passability, sliding, and grazing of periodic motions on the boundary.

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Figures

Figure 8

A simple periodic motion for mapping P87654321: (a) simple symmetry, phase plane, (b) force distribution along displacement, (c) velocity response, and (d) force response. Initial condition (Xi,Ẋi,Ωti)≈(1.0,5.6026,0.0243) for Ω=8.5 (m=5, XL=1, Q0=50, k1=3, k2=5, r=5, V=2, and u=0.6).

Figure 9

A mechanical model for gear transmission without impact

Figure 7

A simple nonstick periodic motion for mapping P875431: (a) phase plane, (b) force distribution along displacement, (c) velocity response, and (d) force response. Initial condition (Xi,Ẋi,Ωti)≈(1.0,12.1694,5.4769) for Ω=8.5 (m=1, XL=1, Q0=100, k1=50, k2=100, r=0.05, V=2, and u=0.35).

Figure 6

Analytical prediction of periodic motion with mapping P875431: (a) switching phase, (b) switching displacement, and (c) switching velocity varying with excitation frequency (m=1, Xc=1, Q0=100, k1=50, k2=100, r=0.05, V=2, and u=0.35)

Figure 5

Mapping structures for a periodic motion

Figure 4

Switching planes and basic mapping in the phase plane: (a) global mapping and (b) local and stick mappings

Figure 3

Subdomains and boundaries

Figure 2

The friction forces for a mechanical model

Figure 1

A mechanical model for mechanical systems with a switching spring and friction

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