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RESEARCH PAPERS

Nonlinear Hybrid-Mode Resonant Forced Oscillations of Sagged Inclined Cables at Avoidances

[+] Author and Article Information
Giuseppe Rega

Department of Structural and Geotechnical Engineering, SAPIENZA University of Rome, via A. Gramsci 53, Rome 00197, Italygiuseppe.rega@uniroma1.it

Narakorn Srinil

Department of Structural and Geotechnical Engineering, SAPIENZA University of Rome, via A. Gramsci 53, Rome 00197, Italynarakorn.srinil@uniroma1.it

J. Comput. Nonlinear Dynam 2(4), 324-336 (Mar 28, 2007) (13 pages) doi:10.1115/1.2756064 History: Received September 15, 2006; Revised March 28, 2007

We investigate nonlinear forced oscillations of sagged inclined cables under planar 1:1 internal resonance at avoidance. To account for frequency avoidance phenomena and associated hybrid modes, actually distinguishing inclined cables from horizontal cables, asymmetric inclined static configurations are considered. Emphasis is placed on highlighting nearly tuned 1:1 resonant interactions involving coupled hybrid modes. The inclined cable is subjected to a uniformly distributed vertical harmonic excitation at primary resonance of a high-frequency mode. Approximate nonlinear partial-differential equations of motion, capturing overall displacement coupling and dynamic extensibility effect, are analytically solved based on a multimode discretization and a second-order multiple scale approach. Bifurcation analyses of both equilibrium and dynamic solutions are carried out via a continuation technique, highlighting the influence of system parameters on internally resonant forced dynamics of avoidance cables. Direct numerical integrations of modulation equations are also performed to validate the continuation prediction and characterize nonlinear coupled dynamics in post-bifurcation states. Depending on the elasto-geometric (cable sag and inclination) and control parameters, and on assigned initial conditions, the hybrid modal interactions undergo several kinds of bifurcations and nonlinear phenomena, along with meaningful transition from periodic to quasiperiodic and chaotic responses. Moreover, corresponding spatio-temporal distributions of cable nonlinear dynamic displacement and tension are manifested.

Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Schematic model of a sagged inclined cable. (b) Planar frequency spectrum and avoidance phenomena of inclined cable with θ=30deg.

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Figure 2

Low- and high-frequency (r,s) modal shapes of u and v displacements: Solid (dashed, dotted) lines denote θ=30deg and λ∕π≈2 (θ=45deg and λ∕π≈2, θ=30deg and λ∕π≈4)

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Figure 3

Percent modal contributions to second-order quadratic coefficients embedded in Krr, Kss, Krs, K1, K2, and K3: (a) θ=30deg, λ∕π≈2; (b) θ=45deg, λ∕π≈2; (c) θ=30deg, λ∕π≈4

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Figure 4

Frequency-response diagrams and bifurcations of inclined cables with θ=30deg, σ=0.04, F=0.002: (a),(b) λ∕π≈2; (c),(d) λ∕π≈2 but with Υr=0; (e),(f) λ∕π≈4

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Figure 5

Frequency-response diagrams and bifurcations of inclined cables with θ=45deg, σ=0.04, F=0.002: (a),(b) fixed-point solution; (c),(d) dynamic solution

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Figure 6

Force-response diagrams and bifurcations of inclined cables with σf=0: Influence of θ and σ

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Figure 7

Dynamic solutions and bifurcations along force-response diagrams of inclined cable with θ=30deg, σf=0: Triangles (circles) denote σ=0.04(σ=0.08)

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Figure 8

Phase portraits showing the transition from periodically to chaotically modulated dynamic response and the boundary crisis of inclined cable in Fig. 5 (from HF2)

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Figure 9

Phase portraits and Poincaré map showing chaotically funnel-shaped dynamic response of inclined cable in Fig. 5 with σf=−0.004812 (from HF1)

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Figure 10

Time response histories showing the intermittently chaotic response of inclined cable in Fig. 5 with σf=0.0035 (from HF3)

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Figure 11

Time response histories and Poincaré map showing the quasiperiodic response of inclined cable in Fig. 5 with σf=0.006 (from HF3)

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Figure 12

PSDs of qs-response histories corresponding to Figs. 8 (a), 8 (b), 9 (c), and 1 (d)

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Figure 13

Phase portraits showing the transition from periodically to chaotically modulated dynamic response of inclined cable in Fig. 7 with σ=0.08

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Figure 14

The effect of initial conditions (set I versus II) on dynamic responses in terms of phase portraits ((a) versus (b)) and Poincaré maps ((c) versus (d)) of inclined cable in Fig. 7 with σ=0.08 and F=0.007

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Figure 15

PSDs of qs-response histories: (a) ((b))corresponding to Figs.  14141414).

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Figure 16

A comparison of space-time varying coupled dynamic (u,v) configurations and tension distributions (Td∕TH), based on fixed-point amplitudes with F=0.002, σ=0.04, and σf=−0.02, over a half period of forced oscillation: corresponding to B1 ((a)–(c)) in Figs.  44(θ=30deg), B1 ((d) and (f)) and B2 ((g)–(i)) in Figs.  55(θ=45deg), respectively.

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Figure 17

A comparison of space-time varying coupled dynamic (u,v) configurations and tension distributions (Td∕TH) based on fixed-point ((a)–(c)) and time-varying ((d)–(f)) amplitudes with θ=45deg, F=0.002, σ=0.04 and σf=0.0225, over a half period of forced oscillation: corresponding to B1 ((a)–(c)) in Figs.  55 and Fig. 8 with t≈211.2–216.2 ((d)–(f)), respectively.

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