Elastodynamics of Nonshallow Suspended Cables: Linear Modal Properties

[+] Author and Article Information
Walter Lacarbonara1

Dipartimento di Ingegneria Strutturale e Geotecnica, University of Rome La Sapienza, via Eudossiana, 18 Rome, 00184 Italy

Achille Paolone, Fabrizio Vestroni

Dipartimento di Ingegneria Strutturale e Geotecnica, University of Rome La Sapienza, via Eudossiana, 18 Rome, 00184 Italy


Author to whom all correspondence should be addressed.

J. Vib. Acoust 129(4), 425-433 (Feb 07, 2007) (9 pages) doi:10.1115/1.2748463 History: Received December 10, 2005; Revised February 07, 2007

A mechanical model describing finite motions of nonshallow cables around the initial catenary configurations is proposed. An exact kinematic formulation accounting for finite displacements is adopted, whereas the material is assumed to be linearly elastic. The nondimensional mechanical parameters governing the motions of nonshallow cables are obtained via a suitable nondimensionalization, and the regions of their physically plausible values are portrayed. The spectral properties of linear unforced undamped vibrations around the initial static configurations are investigated via a Galerkin-Ritz discretization. A classification of the modes is obtained on the basis of their associated energy content, leading to geometric modes, elastostatic modes (with prevalent transverse motions and appreciable stretching), and elastodynamic modes (with prevalent longitudinal motion). Moreover, an extension of Irvine’s model to moderately nonshallow cables is proposed to determine the frequencies and mode shapes in closed form.

Copyright © 2007 by American Society of Mechanical Engineers
Topics: Cables
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Figure 1

The geometry of the cable model: C0 and C indicate the initial static configuration (catenary) and the current configuration, respectively

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Figure 2

(a) Variation of γ with the sag-to-span ratio d and (b) region of admissible physical nondimensional parameters in the γ-λ plane. The curves indicate iso-k (isostiffness) curves: k1=2.5×102, k2=5×103, and k3=5×104. In (b), the curves denoted St, Co, and Ny represent cables of span ℓ=200m and made, respectively, of steel (ρ=7.85×103kg∕m3, E=210GPa), copper (ρ=8.9×103kg∕m3, E=117GPa), and nylon (ρ=1.425×103kg∕m3, E=2.5GPa).

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Figure 3

Variation of the lowest nine natural frequencies with λ∕π obtained with Irvine’s theory (dashed lines) and with the nonshallow theory when γ=0.75 (solid lines)

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Figure 4

Variation of the natural frequencies with λ∕π when (a) γ=1.5 and (b) γ=2.5

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Figure 5

(a) Variation of the natural frequencies with λ∕π when γ=0.75, (b) ratio of the elastic modal energy to the geometric modal energy (in log scale), and (c) ratio of the modal longitudinal kinetic energy to the total kinetic energy

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Figure 6

Close up, around the lowest two elastostatic crossovers, of the loci of the frequencies of modes 7–10 when γ=1.5 along with the mode shapes when λ=8.75π and λ=10.5π

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Figure 7

(Top) Close up, around two elastodynamic crossovers, of the loci of the frequencies of modes 16–20 when γ=1.5. (Bottom) the cable mode shapes when λ=6.81π, (a–d) λ=7.20π (e–h), and λ=7.60π (i–l).

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Figure 8

Modal displacements (left) and strains (right) of the modes at the elastostatic, first and second elastodynamic crossovers when γ=1.5 and (a, b) n=3 and λ=3.74π, (c, d) n=9 and λ=3.583π, and (e, f) n=18 and λ=3.5π. The dashed and solid lines (a, c, e) indicate the longitudinal and transverse components, respectively. The dashed and thin lines (b, d, f) indicate the longitudinal and transverse contributions, respectively, to the modal strain (solid line).



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