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Research Papers

Parametric Resonance of Hopf Bifurcation in a Generalized Beck’s Column

[+] Author and Article Information
Achille Paolone

Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italyachille.paolone@uniroma1.it

Francesco Romeo1

Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza”, Via Gramsci 53, 00197 Roma, Italyfrancesco.romeo@uniroma1.it

Marcello Vasta

Dipartimento di PRICOS, Università “G. D’Annunzio” di Chieti-Pescara, Viale Pindaro 42, 65127 Pescara, Italymvasta@unich.it

1

Corresponding author.

J. Comput. Nonlinear Dynam 4(1), 011003 (Nov 11, 2008) (8 pages) doi:10.1115/1.3007905 History: Received July 25, 2007; Revised March 26, 2008; Published November 11, 2008

A generalized damped Beck’s column under pulsating actions is considered. The nonlinear partial integrodifferential equations of motion and the associated boundary conditions, expanded up to cubic terms, are tackled through a perturbation approach. The multiple scales method is applied to the continuous model in order to obtain the bifurcation equations in the neighborhood of a Hopf bifurcation point in primary parametric resonance. This codimension-2 bifurcation entails two control variables, namely, the amplitude of the static and dynamic components of the follower force, playing the role of detuning and bifurcation parameters, respectively. In the postcritical analysis bifurcation diagrams and relevant phase portraits are examined. Two bifurcation paths associated with specific values of the follower force static component are discussed and the birth of new stable period-2 subharmonic motion is observed.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic view of the model: (a) beam model and (b) kinematic description

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Figure 2

Buckling and Hopf boundaries for the autonomous case

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Figure 3

Regions with different numbers of admissible roots; bifurcation paths at (a) η=−0.001 and (b) η=0.001

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Figure 4

Bifurcation surfaces for (a) negative η and (b) positive η

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Figure 5

Bifurcation diagrams along the paths shown in Fig. 3: (a) η=−0.001 and (b) η=0.001

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Figure 6

Phase portraits in Cartesian coordinates for path a shown in Fig. 3: (a) μ=0.0, region I and (b) μ=0.008, region II

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Figure 7

Phase portraits in Cartesian coordinates for path b shown in Fig. 3: (a) μ=0.0, region IV; (b) μ=0.001, region III; and (c) μ=0.004, region II

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