Research Papers

Competing Dynamic Solutions in a Parametrically Excited Pendulum: Attractor Robustness and Basin Integrity

[+] Author and Article Information
Stefano Lenci

Dipartimento di Architettura, Costruzioni e Strutture, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italylenci@univpm.it

Giuseppe Rega

Dipartimento di Ingegneria Strutturale Geotecnica, Università di Roma “La Sapienza,” Via Antonio Gramsci 53, 00197 Roma, Italygiuseppe.rega@uniroma1.it

J. Comput. Nonlinear Dynam 3(4), 041010 (Sep 02, 2008) (9 pages) doi:10.1115/1.2960468 History: Received May 21, 2007; Revised February 18, 2008; Published September 02, 2008

The competing solutions of a planar pendulum parametrically excited by the vertical motion of the pivot are investigated in terms of both attractor robustness and basin integrity. Two different measures are considered to highlight how the integrity of the system is modified by changing the amplitude of the excitation. Various attractors, both in-well and out-of-well, are considered, and the integrity profiles of each of them are determined. A detailed discussion of the interaction and mutual erosion of the various attractors is performed by clarifying the role of the two complementary measures, and the most relevant characteristics of the erosion profiles are interpreted in terms of the underlying topological mechanisms involving local or global bifurcations.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

The parametric pendulum

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Figure 2

Conservative phase portrait

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Figure 3

Bifurcation diagram of the attractors. HOM1, HOM2, and HET are the thresholds for HOM bifurcation of HS, HOM bifurcation of DR1, and HET bifurcation of DR1 and Ir, respectively.

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Figure 4

The homoclinic bifurcation of DR1 at p=0.655. Only the manifolds of the upper DR1 (rotating in clockwise sense) are reported for a better graphical understanding.

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Figure 5

Basins of attractions for p=0.65. The circles are involved in the definition of the IF.

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Figure 6

Basins of attractions for p=0.7

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Figure 7

Basins of attractions for p=0.90

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Figure 8

Basins of attractions for p=0.96

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Figure 9

Integrity profiles for ω=2

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Figure 10

Basins of attractions for p=1.10

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Figure 11

Basins of attractions for p=1.80

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Figure 12

Integrity profiles for ω=1.8

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Figure 13

Integrity profiles for ω=2.2




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