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Research Papers

A Model for the Wave Localization in a Disordered Periodic Elevated Railway Undergoing In-Plane Vibration

[+] Author and Article Information
Qing-Xu Fu

e-mail: fqxmaster@yahoo.cn

Jian-Fei Lu

e-mail: ljfdoctor@yahoo.com
Department of Civil Engineering,
Jiangsu University,
Zhenjiang, Jiangsu 212013, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 9, 2013; final manuscript received May 21, 2013; published online June 19, 2013. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 135(5), 051027 (Jun 19, 2013) (12 pages) Paper No: VIB-13-1043; doi: 10.1115/1.4024698 History: Received February 09, 2013; Revised May 21, 2013

In this study, a model for the analysis of the wave localization in a special kind of simply-supported beam bridge, namely, the periodic elevated railway (PER), is developed. For simplicity, each span of the PER is supposed to be composed of two longitudinal beams, a pier, and three linking springs. The standard linear solid model is employed to describe the damping of the materials of the piers and beams. Transfer matrix for each span of the PER undergoing in-plane vibration is derived, whereby the wave transfer matrix for each span is obtained. By means of the Wolf's algorithm and using the aforementioned wave transfer matrices, the localization factors accounting for wave localization in the PER are determined. With the proposed model, the influence of the disorder of the beam lengths on the wave localization in the PER is examined. Also, the interactive effect of the damping and the beam-length disorder on the wave localization in the PER is investigated. As a special case, the wave localization in a PER with rigid beam-beam-pier (BBP) junctions is also discussed in this study. Moreover, by the wave transfer matrix method, the wave localization and conversion phenomena in a finite disordered PER segment are investigated. Finally, the relation between the response of a finite disordered PER segment to external loadings and the degrees of the disorder of the PER segment is examined.

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Figures

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Fig. 1

A schematic illustration of a periodic elevated railway rigidly supported on a half-space soil

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Fig. 2

The sign convention for the internal forces of the piers and beams: (a) the sign convention for the internal forces of the piers and (b) the sign convention for the internal forces of the beams

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Fig. 3

The illustration for the n-th BBP junction linking the n-th pier and n-th left as well as right beams undergoing in-plane vibration: (a) the illustration for the overall n-th BBP junction, (b) the illustration for the end of the left beam, and (c) the illustration for the end of the right beam

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Fig. 4

Comparison of the three positive Lyapunov exponents with the imaginary parts of the wavenumbers for the three characteristic waves of the periodic elevated railway: (a) the comparison for the first wave, (b) the comparison for the second wave, and (c) the comparison for the third wave

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Fig. 5

The localization factors versus frequency for the disorder occurring in the beam lengths with the variation coefficient (δ) equal to 0.0, 0.05, and 0.10, respectively: (a) the localization factors for the PER with elastic BBP junctions and (b) the localization factors for the PER with rigid BBP junctions

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Fig. 6

The localization factors versus frequency for the undamped ordered, undamped disordered, damped ordered, and damped disordered PERs with the disorder occurring in the beam lengths of the PER: (a) the localization factors for the PER with elastic BBP junctions, and (b) the localization factors for the PER with rigid BBP junctions

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Fig. 7

Natural logarithm of the magnitudes of the three transmitted characteristic waves in the disordered 300 railway spans with the disorder occurring in the beam lengths of the PER (δ=0.05) when three characteristic waves with the frequency equal to f=16Hz are incident: (a) the type-I wave incident, (b) the type-II wave incident, and (c) the type-III wave incident

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Fig. 8

Natural logarithm of the magnitudes of the three transmitted characteristic waves in the disordered 300 railway spans with the disorder occurring in the beam lengths of the PER (δ=0.05) when three characteristic waves with the frequency equal to f=20Hz are incident: (a) the type-I wave incident, (b) the type-II wave incident, and (c) the type-III wave incident

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Fig. 9

The transverse displacement of the beam section at the left end of the 0-th span of an ordered PER due to a spatially harmonic wave propagating in the half-space soil

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Fig. 10

The transverse displacement of the beam sections at the left end of each span of the 300 disordered railway spans due to the spatially harmonic wave for two frequencies when the variation coefficient of the beam-length disorder is equal to δ=0.0,0.05 and 0.10, respectively: (a) the frequency f=13.25Hz (the resonance frequency) and (b) the frequency f=16Hz (nonresonance frequency)

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