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RESEARCH PAPERS

Response Scenario and Nonsmooth Features in the Nonlinear Dynamics of an Impacting Inverted Pendulum

[+] Author and Article Information
Stefano Lenci

 Dipartimento di Architettura, Costruzioni e StruttureLenci@univpm.it

Lucio Demeio

Dipartimento di Scienze Matematiche,  Università Politecnica delle Marche, Ancona, ItalyL.Demeio@univpm.it

Milena Petrini

Dipartimento di Scienze Matematiche,  Università Politecnica delle Marche, Ancona, ItalyM.Petrini@univpm.it

J. Comput. Nonlinear Dynam 1(1), 56-64 (May 04, 2005) (9 pages) doi:10.1115/1.1944734 History: Revised May 04, 2005

In this work, we perform a systematic numerical investigation of the nonlinear dynamics of an inverted pendulum between lateral rebounding barriers. Three different families of considerably variable attractors—periodic, chaotic, and rest positions with subsequent chattering—are found. All of them are investigated, in detail, and the response scenario is determined by both bifurcation diagrams and behavior charts of single attractors, and overall maps. Attention is focused on local and global bifurcations that lead to the attractor-basin metamorphoses. Numerical results show the extreme richness of the dynamical response of the system, which is deemed to be of interest also in view of prospective mechanical applications.

Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Impacting inverted pendulum

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Figure 2

Phase-space portrait for the conservative system

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Figure 3

Three confined periodic attractors for ω=5 and γ=0.5: (a) phase portraits and (b) time histories

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Figure 4

Three scattered periodic attractors for ω=1.5 and γ=0.4: (a) phase portraits and (b) time histories

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Figure 5

Typical bifurcation diagram for confined attractors, ω=5.0

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Figure 6

Typical bifurcation diagram for scattered attractors, ω=1.5

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Figure 7

Behavior chart for (a) a 1−1−c attractor and (b) a 1−2−s attractor

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Figure 8

Map of coexisting periodic attractors

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Figure 9

Basins of attraction for ω=4.4 and γ=0.61

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Figure 10

Chattering oscillations: Δx(t)≡(1+x(t))104 as a function of t, for ω=5 and (a) γ=1.02, (b) γ=1.04, (c) γ=1.06, and (d) γ=1.08

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Figure 11

The ratio Δti+1∕Δti as a function of the impact sequential number i for ω=5

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Figure 12

Function τ(γ) and its asymptotic approximation for ω=5

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Figure 13

Bifurcation diagrams for (a) ω=3 and (b) ω=18

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Figure 14

Map of robust chaotic attractors

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Figure 15

Scattered chaotic attractor just before the boundary crisis at γ=1.222 and ω=3

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Figure 16

Stable and unstable manifolds of the saddle experiencing homoclinic bifurcation responsible for the boundary crisis at γ=1.22 and ω=3

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Figure 17

Successive enlargements of the bifurcation diagram ensuing from the 1−1−c periodic attractor at ω=12 across its final crisis

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Figure 18

(a) Confined (γ=1.65) and (b) scattered (γ=1.8) chaotic attractors across the homoclinic bifurcation threshold of the hilltop saddle: γ=γcrh=1.656 and ω=18

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Figure 19

Stable and unstable manifolds of the hill top saddle experiencing homoclinic bifurcation responsible for the symmetric boundary crises at γcrh=1.656 and ω=18.

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