Nonlinear vibrations of suspended cables—Part II: Deterministic phenomena

[+] Author and Article Information
Giuseppe Rega

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

Appl. Mech. Rev 57(6), 479-514 (Feb 16, 2005) (36 pages) doi:10.1115/1.1777225 History: Online February 16, 2005
Copyright © 2004 by ASME
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Basins of attraction in the 1/2-subharmonic range (Ω=1.705) with: a) manifolds of direct saddle D1 and b) chaotic attractor 46
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Intricate basins of attraction in the 1/3-subharmonic range: lower P-1 (white core), upper P-1 (outer grey) b), P-3 (inner dark grey), chaos a) or P-6b) (inner grey) 47
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Global mechanism of onset of six-piece chaotic attractor 125
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Interior switching-crisis associated with heteroclinic tangle and involving merging of bands of a chaotic attractor
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Interior bursting-crisis associated with homoclinic tangle and involving widening of a narrow chaotic attractor
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Competing and/or interchanging stable (thick line) response classes in a frequency-response diagram at primary resonance. Perturbation (curves) and numerical (points) results.
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Evolution and/or coexistence of classes of motion with increasing excitation amplitude (1/2-subharmonic resonance)
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In-plane (first column) and out-of-plane (second column) dynamic spatial shapes of the period-two a1 a3 response with direct (solid) and discretized (dashed) approach 95
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Response chart of experimental cable/mass near: a) primary resonance (in-phase support motion) 109 and b) 1/2-subharmonic resonance (out-of-phase support motion) 105
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Competition of classes of regular motion in overlapping regions of a response chart 90
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Transitions between couples of classes of motion: full orbits in the horizontal-vertical plane, transient and steady Poincaré points 122
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Frequency-response diagram at primary resonance (in-phase support motion) for: a) slacker and b) crossover cable 90
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Qualitative characterization of chaotic response under: a) in-phase and b) out-of-phase support motion 125
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Typical steps for quantitative characterization of chaotic response in Fig. 32a125
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Schematic bifurcation diagram (in-phase support motion, primary resonance). H: Hopf, PL: phase-locking, TB: torus breakdown, SC: saddle cycle. The highlighted motion classes refer to periodic (Pm[-M3]), quasiperiodic (nT-QPm[-M3]) and chaotic (CH1,CH2) attractors 109.
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Experimental eigenfunctions relevant to: a) periodic (P1:V1),b) quasiperiodic (2T-QP1:H1,V1), and c) chaotic (CH2:V1,H1,H2) attractors. Σ, percentage of signal power.
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Poincaré sections of configuration space in the neighborhood of homoclinic bifurcation 110
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Nonstationary three-mode response in the APMEs of 4-dof theoretical model 125
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Periodic amplitudes a), periodically amplitude modulated motion b), and frequency spectrum c) of the q2 dof 125
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Motion on a: a) two-torus and b) seemingly chaotic attractor in the ODEs 125
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Frequency-amplitude relationship for different cables: perturbation solution 15
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Frequency-amplitude relationship for different cables: numerical simulations 16
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Effective nonlinearity coefficient of the backbone curve for different discretized models
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Undamped frequency-response curves for: a) a nearly taut string and b) a suspended cable 30
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State plane for damped response of the cable (σ=−0.15, p=0.05, μ=0.02) 30
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Frequency-response curves of order-3 superharmonic oscillations for various cables (p=0.009, μ=0.04). Dashed lines denote unstable solutions 32.
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Frequency-response curves in the superharmonic range: harmonic balance solution (continuous lines: thick, stable; thin, unstable) and computer simulation results (lines with large marks) 33
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a) Regions of existence (II,III) or non-existence (I) and b) frequency-response curve of 1/2-subharmonic oscillations for a nearly taut cable (thick) and a sagged cable (thin) (μ=0.1) 34
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a) Regions of existence (II,III) or non-existence (I) and b) frequency-response curve of 1/3-subharmonic oscillations for a nearly taut cable (thick) and a sagged cable (thin) (μ=0.1) 34
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Frequency-response curves in the: a) 1/2- and b) 1/3-subharmonic range: harmonic balance approximations 35
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Frequency-response curves in the: a) 1/2- and b) 1/3-subharmonic range: continuation of fixed points of Poincaré map 35
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Periodic and chaotic zones of response in control space (P,Ω)37
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Bifurcation diagram in the 1/2-subharmonic range in terms of: a) periodicity and b) Lyapunov exponents (P=0.04)35
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Bifurcation diagram in the 1/3-subharmonic range in terms of: a) periodicity and b) Lyapunov exponents (P=0.40)35
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Frequency-response curves in the: a) 1/2- and b) 1/3-subharmonic range: harmonic balance and computer simulation results 33
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Bifurcation predictive capability of stability numerical analysis of low-order harmonic balance solutions. Comparison with computer simulation results (zero ic) 33.
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Chaotic attractor in the: a) 1/2-subharmonic, b) 1/3-subharmonic, and c) superharmonic ranges 35
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Bifurcation diagram with main sink and saddle paths in the 1/2-subharmonic range 46
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Bifurcation diagram (one-third) with main sink and saddle paths in the 1/3-subharmonic range 47




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