Research Papers

Influence of Modal Coupling on the Nonlinear Dynamics of Augusti’s Model

[+] Author and Article Information
Diego Orlando

TecGraf -Computer Graphics Technology Group, Pontifical Catholic University of Rio de Janeiro, PUC-Rio Rio de Janeiro, RJ 22451-900, Brazildorlando@tecgraf.puc-rio.br

Paulo B. Gonçalves

Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro, PUC-Rio Rio de Janeiro, RJ 22451-900, Brazilpaulo@puc-rio.br

Giuseppe Rega

Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza Università di Roma, Rome 00197, Italygiuseppe.rega@uniroma1.it

Stefano Lenci

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

J. Comput. Nonlinear Dynam 6(4), 041014 (Jun 14, 2011) (11 pages) doi:10.1115/1.4003880 History: Received June 07, 2010; Revised March 18, 2011; Published June 14, 2011; Online June 14, 2011

The nonlinear behavior and stability under static and dynamic loads of an inverted spatial pendulum with rotational springs in two perpendicular planes, called Augusti’s model, is analyzed in this paper. This 2DOF lumped-parameter system is an archetypal model of modal interaction in stability theory representing a large class of structural problems. When the system displays coincident buckling loads, several post-buckling paths emerge from the bifurcation point (critical load) along the fundamental path. This leads to a complex potential energy surface. Herein, we aim to investigate the influence of nonlinear modal interactions on the dynamic behavior of Augusti’s model. Coupled/uncoupled dynamic responses, bifurcations, escape from the pre-buckling potential well, stability, and space-time-varying displacements; attractor-manifold-basin phase portraits are numerically evaluated with the aim of enlightening the system complex response. The investigation of basins evolution due to variation of system parameters leads to the determination of erosion profiles and integrity measures which enlighten the loss of safety of the structure due to penetration of eroding fractal tongues into the safe basin.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

Augusti’s two-degree-of-freedom model: (a) Augusti’s model and (b) deformed configuration

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Figure 2

Equilibrium paths of the system

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Figure 3

Curves of equal potential energy for λ=0.9. PS: saddles. PMi: stable position corresponding to a local minimum (pre-buckling configuration).

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Figure 4

Four two-dimensional sections (phase plots) of the safe pre-buckling region. λ=0.9 and ωp=1. (a) Plane θ1×θ2, (b) plane θ1×θ̇1, (c) plane θ2×θ̇2, and (d) plane θ̇1×θ̇2

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Figure 5

The stable and unstable manifolds of the four saddles that surround the safe pre-buckling region. λ=0.9 and ωp=1. (a) Plane θ1×θ2, (b) plane θ1×θ̇1, (c) plane θ2×θ̇2, and (d) plane θ̇1×θ̇2

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Figure 6

Coupled and uncoupled stability boundaries in excitation control space for the forcing direction φ=0 deg. Fesc: escape load.

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Figure 7

Stability boundaries in force control space for different values of the forcing direction φ

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Figure 8

Mapping of the local bifurcations in the fundamental resonance region prior to escape for φ=45 deg

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Figure 9

Bifurcation diagrams for φ=45 deg, illustrating the local bifurcations for two excitation frequencies. SN: saddle-node; PS: supercritical pitchfork; PSb: subcritical pitchfork; FS: supercritical flip; E: escape. (a) Ω=1/3 and (b) Ω=0.4.

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Figure 10

Bifurcation diagrams for φ=0 deg and Ω=1/3. Coupled and uncoupled cases. PS: supercritical pitchfork bifurcation. H: Hopf bifurcation. E: escape. (a) Coupled and (b) uncoupled.

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Figure 11

Comparison of the basin of attraction of the coupled and uncoupled cases for φ=0 deg, F=0.10, and Ω=1/3. (a) Coupled and (b) uncoupled.

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Figure 12

Bifurcation diagrams for two values of the forcing direction φ for Ω=1. (a) φ=0 deg-uncoupled and (b) φ=2 deg -coupled

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Figure 13

Behavior of uncoupled system for φ=0 deg, F=0.74, and Ω=1. (a) Phase portrait and (b) basin of attraction.

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Figure 14

Phase portrait for φ=2 deg, F=0.67, and Ω=1. (a) Plane θ1×θ̇1 and (b) plane θ2×θ̇2.

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Figure 15

Basin of attraction with increasing forcing magnitude F for φ=2 deg and Ω=1. (a) F=0.01-plane θ1×θ2 and (b) F=0.65-plane θ1×θ̇1.

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Figure 16

Variation of the escape load with the forcing direction φ for the resonance regions

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Figure 17

Erosion of the basin of attraction with increasing forcing magnitude F for φ=45 deg and Ω=1/3. (a) F=0.01 and (b) F=0.10.

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Figure 18

Variation of the integrity factor IF as a function of the forcing magnitude F for two values of the forcing direction φ in the fundamental resonance region, Ω=1/3




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